Menghua
Zhao
^{ab},
François
Lequeux
^{b},
Tetsuharu
Narita
^{b},
Matthieu
Roché
^{a},
Laurent
Limat
^{a} and
Julien
Dervaux
*^{a}
^{a}Laboratoire Matière et Systèmes Complexes, CNRS UMR 7057, Universitée Denis Diderot, 10 Rue A. Domon et L. Duquet, Paris, France. E-mail: julien.dervaux@univ-paris-diderot.fr
^{b}Sciences et Ingénierie de la Matière Molle, CNRS UMR 7615, Ecole Supérieure de Physique et de Chimie Industrielles de la Ville de Paris (ESPCI) ParisTech, PSL University, 10 Rue Vauquelin, Paris, France

Received
31st August 2017
, Accepted 31st October 2017

First published on 31st October 2017

Elastocapillarity describes the deformations of soft materials by surface tensions and is involved in a broad range of applications, from microelectromechanical devices to cell patterning on soft surfaces. Although the vast majority of elastocapillarity experiments are performed on soft gels, because of their tunable mechanical properties, the theoretical interpretation of these data has been so far undertaken solely within the framework of linear elasticity, neglecting the porous nature of gels. We investigate in this work the deformation of a thick poroelastic layer with surface tension subjected to an arbitrary distribution of time-dependent axisymmetric surface forces. Following the derivation of a general analytical solution, we then focus on the specific problem of a liquid drop sitting on a soft poroelastic substrate. We investigate how the deformation and the solvent concentration field evolve in time for various droplet sizes. In particular, we show that the ridge height beneath the triple line grows logarithmically in time as the liquid migrates toward the ridge. We then study the relaxation of the ridge following the removal of the drop and show that the drop leaves long-lived footprints after removal which may affect surface and wetting properties of gel layers and also the motion of living cells on soft materials. Preliminary experiments performed with water droplets on soft PDMS gel layers are in excellent agreement with the theoretical predictions.

Despite these important advances, and while many experimental studies are performed on gels, most theoretical studies on elastowetting have assumed that the soft deformable substrates are purely (linear) elastic materials. It was only recently recognized that even some simple features of static elastowetting, such as the formation of the ridge beneath the triple line below a sessile drop,^{21} are in fact time dependent processes. This non-instantaneous response differs from that of a purely elastic solid, but the rate-limiting mechanism could be either network rearrangement (viscoelasticity), solvent diffusion (poroelasticity) or a combination of both. However, intriguing results, such as the coexistence of multiple phases at the contact line in recent indentation experiments,^{22} unambiguously highlight the need to use a multiphase model (such as the poroelastic theory) to rationalize these observations.

In the simplest case, a gel can undergo two modes of deformation.^{26} On a very short timescale following the sudden application of a force on a gel sample, the solvent molecules do not have time to diffuse and the gel behaves as an incompressible elastic solid while hydrostatic pressure builds up within the liquid phase.^{26,27} On this short timescale, the gel can change its shape but not its volume. In reaction to this pressure however, the solvent molecules migrate. This long-range migration process occurs on timescales that depend on the size of the sample and allows the gel to change both its shape and its volume.^{28–31} Therefore, while incompressible on a short timescale, a gel is highly compressible on a long timescale. In the limit of small deformations, this behavior is well described by the theory of linear poroelasticity, initially developed by Biot in 1941^{32} to describe soil consolidation.

In general, however, not all crosslinks in polymer networks are permanent and part of them may be capable of dynamic dissociation and re-association, such as physical gels^{33} or interpenetrating gel networks.^{34,35} These reversible crosslinks, together with the rearrangement of the polymer chains and the viscosity of the solvent itself, endow the gels with additional viscoelastic properties.^{36–38} Because poro- and visco-elastic processes occur simultaneously in gels, their time-dependent mechanical response is in general rather complex^{39–43} and must be described by poro-visco-elastic theory. Because of this interplay, and although several models of poro-visco-elasticity have been developed,^{29,44–47} experimental protocols allowing the extraction of the poro- and visco-elastic material parameters have been devised only recently.^{40,41} While some work has been done to incorporate the viscoelastic response of the gel within the theoretical framework^{23,48} of elastowetting, neither the poro-elastic response nor the poro-visco-elastic response has been incorporated so far within the theoretical framework that describes the time-dependent behavior of soft solids near the triple line.

In order to grasp the new physical effects induced by the poroelasticy of the substrate on elastowetting phenomena, we will neglect visco-elastic behaviors in the present work and we will study the time dependent behavior of a thick poroelastic substrate subjected to an arbitrary, but axisymmetric, time-dependent distribution of normal surface forces. In the next section, we briefly recall the field and constitutive equations of linear poroelasticity. Next, we present a general analytical solution to the poroelastowetting problem. We then investigate the specific case of poroelastic deformation due to the deposition (and subsequent removal) of a hemispherical drop at the surface of the gel. We then discuss our findings and highlight future development as well as outstanding questions of broad scientific interest.

(1) |

(2) |

(3) |

In the framework of linear poroelasticity, the stress tensor σ is given by:

(4) |

Tr(ε) = (c − c_{0})Ω | (5) |

The mechanical equilibrium in the bulk of the poroelastic layer is described by the Navier equations:

(6) |

Combining the equations above we obtain

(7) |

(8) |

(9) |

(10) |

(11) |

In addition, we assume that the substrate is infinitely thick and thus the displacement and stress fields vanish for z → −∞.

(r,z,t) = u(r,z,t)_{r} + v(r,z,t)_{z} | (12) |

Furthermore, we only consider in this paper the case where the traction force at the free boundary is purely normal, i.e. ∝ _{z}. Although we will consider later time-dependent forcing, we first focus on the step response of the system i.e. when the surface force distribution is suddenly applied at t = 0 and subsequently maintained for t ≥ 0. We may therefore write = f_{z}(r)H(t)_{z}, where H(t) is the Heaviside step function. Thus, we have at the free surface:

(13) |

σ_{rz}(r, z = 0, t) = 0 | (14) |

(15) |

(16) |

(17) |

(18) |

(19) |

The above ordinary differential equation is readily solved and involves four unknown coefficients (that depend on s). Two of them are cancelled to satisfy . The two remaining functions are found by making use of the two boundary conditions (13), (14) and we find:

(20) |

(21) |

(22) |

(23) |

(24) |

(25) |

(26) |

Plugging the above results in (7), we obtain the following non homogeneous fourth-order linear partial differential equation for û:

(27) |

This equation is supplemented by the two boundary conditions (14) and (11) at the free surface while . Furthermore, the initial condition is given by û(s,z,0) = û^{i} where û^{i} is defined by (21). Because we are interested for the first time in the step response of the system, this problem is best solved by introducing the Laplace transform Û(s,z,ω) of û(s,z,t) defined as:

(28) |

Plugging the above expression into the evolution eqn (27) for û(s,z,t) and making use of the initial condition û(s,z,0) = û^{i}, we now obtain a non-homogeneous fourth order linear ordinary differential equation that is easily solved by using the four boundary conditions mentioned previously.

(29) |

(30) |

(31) |

(32) |

The response of the gel is thus that of an incompressible solid at short time while it behaves as a compressible solid at long time. This apparent compressibility is due to the ability of the solvent to migrate and is quantified by the poroelastic Poisson ratio ν. When ν = 1/2, the gel cannot absorb or release any solvent and the final and initial states are identical. In that case, the inverse Laplace transform can be performed analytically and the deflection is (s,t) = ^{i}(s)H(t). This solution is the purely elastic response originally derived by Jerison and later by Style and Dufresnes. The displacement of the interface at large distances (small wavenumber s) behaves as ∼_{z}(s)/(2Gs) and the deflection is thus damped by the bulk elasticity. At small distances (large s), the deflection behaves as ∼_{z}(s)/(s^{2}γ_{s}) and is thus dominated by surface tension. The crossover between the elastic and the capillary regime occurs at the scale of the elastocapillary length _{s}.

(33) |

This general solution will be investigated at the end of the next section but we first turn to the analysis of the solution (30) in the case of a hemispherical droplet on a poroelastic substrate (Fig. 1).

(34) |

With this choice of surface force distribution, the interface profile ζ(r,t) is given by:

(35) |

Let us note here that, in general, the wet and dry parts of the solid are likely to have different surface energies. However, taking this effect into account leads to a discontinuous boundary value problem that can be further transformed into coupled integral equations which cannot be solved analytically. This problem therefore remains, as mentioned in the introduction, an open question. Because we wish to focus this work on the effect of poroelasticity on elastowetting, we will assume here that the surface energies of the dry and wet parts of the solids are equal and given by γ_{s}. We now turn to the detailed analysis of eqn (35) for specific cases of broad physical interest.

(36) |

(37) |

This non-trivial overshooting behavior can be understood by analyzing the two forces that are applied to the surface of the poroelastic substrate. While both forces imply migration of solvent over a lengthscale R, the Laplace pressure in the drop acts as a distributed pressure on the surface that pushes fluid only in the outward radial direction. On the other hand, the traction due to the air/liquid interface is a force localized at the triple line and draws fluid from both the inside and the outside of the drop. As a consequence the increase in height due to this traction relaxes twice as fast as the decrease in height due to the Laplace pressure. The combination of these two forces with slightly different timescales therefore produces the overshoot behavior seen in the inset of Fig. 2C.

Because the inverse Hankel–Laplace cannot be evaluated analytically, it is not possible to provide a simple expression (in the time domain) for the time evolution of the ridge height h(t). However, some crude approximations can be performed in order to gain further insight into the behavior of h(t). In the limit of large drop, we focus on the evolution of h(t) between the two intermediate timescales _{s}^{2}/D* ≪ t ≪ R^{2}/D* and we will make the crude approximation that the evolution of h(t) in this regime is mostly due the evolution of the corresponding lengthscales 1/R ≪ s ≪ 1/_{s} and we will check later that this approximation is self-consistent. In this limit, the Laplace transform of the increase of the ridge height h(t) − h(0^{+}) is then approximately . This simpler expression can then be integrated along s and the resulting expression can finally be inverted in the time domain analytically to yield the scaling

h(t) − h(0^{+}) ∝ _{s}log(tD*/_{s}^{2}) | (38) |

As seen in Fig. 2C, this expression fits rather well the numerical result between the two intermediate timescales _{s}^{2}/D* ≪ t ≪ R^{2}/D*, as expected from our assumptions. Besides providing a reasonable approximation to the evolution of the ridge height, it also shows that the relevant timescale for the evolution of the ridge created by large drops is _{s}^{2}/D*. Beneath the drop, the depth of the valley is, at leading order, independent of the drop size and increases over time, from until it reaches . As seen in Fig. 2B, the formula above are a good approximation for the case R/_{s} = 100. Beneath the drop, the chemical potential increases right after the deposition. We find that, for large drops, the chemical potential beneath the drop at t = 0^{+} is given by:

(39) |

At the contact line however, the chemical potential diverges as log|r − R|. In the final state, the chemical potential relaxes everywhere to μ_{0}. Similarly, the concentration of the solvent, initially equal to c_{0} reaches the following value beneath the drop in the steady state.

(40) |

As can be seen from eqn (39) and (40), although the depth of the valley increases over time, the change in the concentration beneath the drop is very small. On the other hand, and while the change in the amplitude of the ridge is of the same order than that of the depth of the valley, the solvent concentration increases sharply (it also diverges as log|r − R|) beneath the ridge. We therefore only plot here the concentration (Fig. 2D–G) and chemical potential (Fig. 2H–K) fields in the vicinity of the contact line. As seen in those panels, the solvent concentration field (c − c_{0}) is zero at t = 0^{+} but then increases sharply near the contact line where we also notice the radial (inward) displacement of the triple line over time.

(41) |

As the consequence of the lenticular shape, we therefore expect the ridge height to be asymptotically zero at leading order. Right after the deposition, the height suddenly jumps to a small height h(0^{+}) which, for small drops is given by the first non-zero term of an expansion in R/_{s}:

(42) |

After relaxation, the ridge height reaches h(∞) which is given asymptotically by:

(43) |

We note that the ridge height is now quadratic in R/_{s}. The consequence of this is that, at first order in R/_{s}, and in stark contrast with large drops, the profile is flat outside the drop. To the same order the profile is thus independent of time for small drops, as can be seen in Fig. 3B. This effect could in fact be expected since the shape of the substrate deformation is controlled solely by capillarity for small drops. As it is therefore independent of the mechanical properties of the substrate (again, to first order in R/_{s}), the profile is both independent of the shear modulus and the poroelastic Poisson ratio, i.e. of the ability of the substrate to reorganize the solvent. We note that, in contrast with large drops, the ratio h(∞)/h(0^{+}) is smaller than 1 and thus small drops gradually sink inside the gel although this is a second order effect on R/_{s}. This can also be seen in Fig. 3C that shows the ratio h(∞)/h(0^{+}) as a function of the drop size R ≪ _{s} for different values of the poroelastic ratio ν. We can also note from this panel that the transition from the symmetric ridge of large drops to the tilted ridge of small drops is quite broad and occurs over several decades of the ratio R/_{s}. This result is quite similar to the transition observed in purely elastic systems that we investigated previously.^{24} Furthermore, note that since R ≪ _{s}, the relevant timescale for the evolution of the ridge profile is not _{s}^{2}/D* anymore, as was the case for large drops, but R^{2}/D*. Now if the shape of the substrate is, at leading order, independent of time, how does the solvent evolve? Beneath the drop, the chemical potential increases right after the deposition while it drops under the ridge. Asymptotically, we find that the chemical potential beneath the drop is given by:

(44) |

As the chemical potential converges to μ_{0} away from the drop, there is indeed a gradient of chemical potential that drives fluid motion. In the final state, the concentration of the solvent beneath the drop is given by:

(45) |

The corresponding concentration and chemical potential are plotted in Fig. 3D–G and H–K, respectively.

ζ_{res}(r,t) = ζ(r,t) − H(t − τ_{res})ζ(r,t − τ_{res}) | (46) |

ζ_{res}(r,τ_{life}) = h_{c} | (47) |

In the present study, the value of this critical thickness is of course arbitrary but it can be quantified for specific applications. In the context of wetting for example, surface defects as small as 10 nm can pin the contact line and affect the static equilibrium angle. As a consequence, if the footprint of a drop is thicker than this critical thickness, it will have consequences at the macroscopic scale on the wetting properties of the gel for instance. As seen in Fig. 4D, the lifetime of the drop footprint strongly depends on this critical thickness and shows a non-trivial dependence on the residence time τ_{res} of the drop. We first note that the residence time must be larger than a critical value for the height of the footprint to be larger than h_{c}. This effect can be seen in the inset of Fig. 4D. Above this critical residence time, the footprint lifetime τ_{life} first increases with the residence time τ_{res} until the residence time becomes comparable with the equilibrium time R^{2}/D*. After this value, the lifetime decreases. This decrease is simply the consequence of the overshoot effect described previously for the growth of the ridge. When the residence time is much larger than R^{2}/D*, then the gel has reached its equilibrium before the drop is removed. In that case, the lifetime of the footprint does not depend on the residence time of the drop and therefore τ_{life} saturates to a finite value.

Fig. 5 Relaxation of the ridge under the contact line following the removal of the drop for a resting time of 30 minutes (A–C) and 2 minutes (D–F). In both cases 5 μL water droplets were placed on a freshly prepared thick PDMS substrate (thickness 1358 μm, shear modulus 1.2 kPa) and a region of ∼750 × 750 μm^{2} was scanned at regular intervals using a 3D profiler (Microsurf 3D, Fogal Nanotech, France) with a lateral resolution of 1.89 μm and a vertical resolution of 50 nm. The first scan was acquired roughly for 20 s following the removal of the drops. The color-coded heights are in microns. As seen in (D–F), no trace could be detected for a residence time of 2 minutes. For a residence time of 30 minutes on the other hand (A–C), the footprint of the drop is initially around 0.5 μm and is still around 100 nm after 4800 s. (G) Time evolution of the height of the trace following drop removal (same parameters as in A–C). The blue circles are the experimental data and the solid orange curve is the theoretical prediction given by eqn (46). For the theoretical curve, the droplet radius and substrate stiffness were taken from the experiment while the surface tension was taken as γ_{s} = 40 mN m^{−1} and the poroelastic Poisson ratio was taken as ν = 0.3. The effective diffusion coefficient D* was found by fitting the data with the model. The best fit for D* was found to be ∼ 2 × 10^{−11} m^{2} s^{−1}, in good agreement with other values found in the literature. |

In our study, we have also seen that another consequence of merging the linear poroelastic theory with the elastowetting problem introduces a new divergence: the solvent concentration diverges as ∼log|r − R| near the contact line. While several approaches might be able to regularize this divergence, for example by taking into account the finite thickness of the gel, the material and geometrical nonlinearities or through the introduction of a finite width for the contact line, the existence of this divergence suggests that extreme phenomena such as phase separation, fracture or instabilities could occur at the contact line.^{60,61} Indeed, the coexistence of multiple phases at the contact line has been recently reported in indentation experiments.^{22} Further work, for example based on nonlinear poroelastic theory, will be needed in order to shed light on the behavior of gels near contact lines.

In a different line of thought, the wetting of saturated gels by drops of their own solvent opens interesting questions. Because the presence of a drop changes the chemical potential away from the drop, several drops may interact with each other by mass exchange throughout the gel. For thick gels (when the thickness of the gel is much larger than _{s} and R), the chemical potential increases above its reference value away from the drop and will tend to suck fluid inside the gel. Because large drops will create a stronger change in the chemical potential than small drops we thus expect large drops to grow at the expense of smaller droplets. For thinner gels however, the effect of finite depth is likely to form a dimple within which the chemical potential drops below its reference value, and thus promotes the growth of smaller droplets nearby. Although speculative, this possibility might open a new route to new original methods to control droplet nucleation and dew collection on soft materials.

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