# Complexity in wetting dynamics

Matthieu Roché, Laurence Talini, Emilie Verneuil

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Matthieu Roché, Laurence Talini, Emilie Verneuil. Complexity in wetting dynamics. Langmuir, 2024, 2023 Pioneers in Applied and Fundamental Interfacial Chemistry: Janet A. W. Elliott, 40, pp.2830-2848. 10.1021/acs.langmuir.3c03292. hal-04278357

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Complexity in wetting dynamics

Matthieu Roché,†,‡ Laurence Talini,*,¶ and Emilie Verneuil§

†Matière et Systèmes Complexes, Université Paris Cité, CNRS UMR 7057, Paris F-75013

‡Department of Materials Physics, Research School of Physics, Australian National
University, Australia

¶CNRS, Surface du Verre et Interfaces, Saint-Gobain, 93300 Aubervilliers, France

§CNRS Sciences et Ingénierie de la Matière Molle, ESPCI Paris, PSL Research
University, Sorbonne Université, Paris, France

E-mail: laurence.talini@cnrs.fr

Abstract

The spreading dynamics of a droplet of pure liquid deposited on a rigid, non-soluble
substrate has been extensively investigated. In a purely hydrodynamic description, the
dynamics of the contact line is determined by a balance between the energy associated
with the capillary driving force and the energy dissipated by the viscous shear in the
liquid. This balance is expressed by the Cox-Voinov law, which relates the spreading
velocity to the contact angle. More recently, complex situations have been examined
in which dissipation and/or the driving force may be strongly modified, leading to
sometimes spectacular changes in wetting dynamics. We review recent examples of
effects at the origin of deviations from the hydrodynamic model, which may involve
physical or chemical modifications of the substrate or of the wetting liquid, occurring
at scales ranging from the molecular to the mesoscopic.
# 1 Introduction

The seemingly straightforward scenario in which a droplet spreads on a solid substrate still challenges engineers and scientists after decades of studies. Because the scales involved range from the molecular level at the contact line to the scale of the droplet itself, this problem encompasses a range of complexities that demand intricate descriptions and meticulous experimentation. The simplest theoretical description of the dynamics of wetting, known as the Cox-Voinov law, balances the work of the driving capillary force induced by the deformation of the liquid/gas interface away from its equilibrium shape and the energy lost by viscous dissipation in the droplet. This law is a crucial tool in the study of wetting since its successes and failures show how the wetting dynamics depends not only on hydrodynamics but also on the physico-chemistry at play at the contact line. The rich literature documenting both experimental and theoretical studies with the aim of gaining a deeper understanding of these issues over the last fifty years is reviewed in numerous works, notably in.<sup>1-3</sup>

A reason for the sustained interest in wetting dynamics lies in the exploration of more complex situations, beyond that of a pure liquid wetting a rigid substrate. In addition to their fundamental interest, complex conditions are often encountered in industrial applications. For instance, in the wet coating process widely used in industry, complex fluids generally containing solutes or suspended particles are needed to form a continuous film on a substrate at the highest possible speed. The characteristic length and time scales of the fluid can lead to wetting behaviours that significantly depart from purely hydrodynamic phenomena. In other applications, coatings are applied to control the wetting properties of surfaces. As an example, hydrogel layers are used as anti-fogging coatings because they promote the formation of a condensed water film instead of droplets, which alter optical properties.<sup>4,5</sup> This involves the wetting of a simple liquid on a poroelastic and deformable coating. A description of wetting in these conditions has yet to be proposed.

The present review focuses on the experimental aspects of the recent literature on wetting dynamics, with a particular emphasis on the effects related to the physico-chemistry of
the systems under consideration. We begin with the hydrodynamic description of wetting dynamics for simple liquids and solids, and then illustrate by providing a non-exhaustive set of examples how this dynamics can deviate from the Cox-Voinov law when a higher degree of complexity in the system is considered.

The text is organised as follows: In Section 2, we recall how the Cox-Voinov law is established. Then we review some effects that modify either the driving capillary force or the nature of the dissipative process, namely: line friction and retardation mechanisms in Section 3, dissipation in the substrate in Section 4, surfactant-induced Marangoni effects in Section 5. The last section deals with finite-size effects.

## 2 Hydrodynamic model

We recall here the basic principles underlying the hydrodynamic model describing the relation between the shape of the interface of the droplet in the vicinity of the contact line and its velocity. Our aim is not to provide a thorough discussion of the physics at play in this region. We suggest that the reader refers to past reviews¹,³,⁶ for more details.

We consider a droplet of liquid of density $ρ$, viscosity $η$ and surface tension $γ$ placed on a solid substrate. We denote respectively as $γ_{sv}$ and $γ_{sl}$ the surface energies of the solid with air and with the liquid. The droplet is in equilibrium when its contact angle, defined as the angle of the contact line with the solid substrate measured inside the droplet (see figure 1), has the equilibrium value $θ_e$. In partial wetting conditions, $γ_{sv} - γ_{sl} < γ$, and $θ_e$ has a finite value given by the balance of the horizontal forces per unit length acting on the contact line, given by the Young-Dupré law,

$$ \cos \theta_e = \frac{\gamma_{sv} - \gamma_{sl}}{\gamma}. \qquad (1) $$

In total wetting conditions, $γ_{sv} - γ_{sl} > γ$, and the equilibrium angle $θ_e$ is zero. The liquid forms a film with a thickness determined by the inter-molecular forces at stake.
<figure><img src="image_2.png" /><figcaption>Figure 1: Schematics of a droplet sitting on a solid substrate and of the forces acting on the contact line.</figcaption></figure>

When the contact angle has a value different from its equilibrium one, the contact line is submitted to the capillary force

$$F = \gamma_{sv} - \gamma_{sl} - \gamma\cos\theta = \gamma(\cos\theta_e - \cos\theta) \quad (2)$$

The contact line has then a velocity $U$, which can be related to the angle $\theta$ by considering that the work of the driving capillary force is entirely dissipated by the viscous flow generated in the drop. In the picture suggested by de Gennes,¹ the vicinity of the contact line is assimilated to a wedge in which a Poiseuille flow takes place, as shown in figure 2, with average velocity equal to $U$. In this simplified view, the wedge height $h$ at a distance $x$ to the contact line is given by $h = \theta x$. More refined derivations accounting for the curvature of the air liquid interface are not detailed here but ascribed the Poiseuille flow to the gradient of capillary pressure resulting from curvature effects.

If a zero velocity boundary condition at the substrate is assumed, the shear stress scales as $\eta U/h$. Viscous stresses at the contact line should then diverge and prevent the motion of the contact line, in opposition with our everyday experience. This paradox, identified by Huh and Scriven⁷, is solved by considering that effects allowing the line to slide on the solid surface are at play below a characteristic microscopic distance to the contact line, $\ell$. In this frame, the power dissipated by the viscous flow is computed at distances larger than the
cut-off length $ℓ$

$$P_{vis} = \int_{\ell}^{x} dx' \int_{0}^{h} \eta \left( \frac{dv}{dz} \right)^{2} dz \quad (3)$$

with the notations defined in Fig. 2 and where $v$ is the velocity field having a mean value equal to the contact line velocity $U$, and $x$ the distance to the contact line at which the angle $θ$ is measured.

<figure><img src="image_3.png" /><figcaption>Figure 2: Schematized view of a drop spreading on a solid substrate and of the lubrication flow in the vicinity of the contact line, which is assimilated to a wedge with angle $θ$.</figcaption></figure>

Since $dv/dz \propto U/h$ with the wedge height varying as $h = θx'$ with the distance $x'$ to the contact line, the expected scaling for the dissipated power is $P_{vis} \propto \eta U^2 \theta \ln(x/\ell)$. In the limit of small angles, the power of the driving force is $P_{cap} \simeq \gamma U(\theta_e^2 - \theta^2)/2$. The relationship between velocity and contact angle is obtained by equating powers, which yields

$$\theta^3 - \theta_e^3 \propto \frac{\eta U}{\gamma} \ln \left( \frac{x}{\ell} \right). \quad (4)$$

The difference of the cubed dynamic and static angles is therefore proportional to the capillary number, $Ca = \eta U/\gamma$, that compares viscous drag to surface tension forces. Increasing capillary numbers, i.e. increasing energy dissipation in the liquid, result in increasing differences between the dynamic and the equilibrium angles.

A full resolution of Stokes' equation in the lubrication approximation, made indepen-
dently by Voinov⁸ and Cox⁹ yields

$$g(\theta(x)) - g(\theta(\ell)) = Ca \ln \left(\frac{x}{\ell}\right) \qquad (5)$$

where $g(\theta) = \int_0^\theta \frac{\beta - \cos\beta \sin\beta}{\sin\beta} d\beta$. For angles smaller than $3\pi/4$, the function $g$ can be approximated,¹⁰ which yields the relation known as the Cox-Voinov law

$$\theta^3 - \theta_e^3 = 9Ca \ln \left(\frac{x}{\ell}\right) \qquad (6)$$

which is close to that given by the scaling analysis (Eq. 4), but remains valid even for large values of the dynamic angle.

Cox has generalised the result to cases in which the droplet is immersed in a viscous liquid instead of air.⁹ In this frame, the relation between angle and velocity can be approximated by¹¹

$$\theta^a - \theta_e^a = bCa \ln \left(\frac{x}{\ell}\right) \qquad (7)$$

where $a$ and $b$ depend on the viscosity ratio of the drop and surrounding liquid.

Equations 6 and 7 are valid at distances to the contact line where capillary prevails over gravity, i.e. smaller than the capillary length, $\ell_c = \sqrt{\gamma/(\rho g)}$, where $g$ is the acceleration of gravity. As a result, the radius variations with time of viscous droplets with radii $R < \ell_c$ can be derived from Eq. 6. For a zero equilibrium angle, $U = \dot{R} \propto \theta^3$ and conservation of droplet volume imposes that $R^3 \propto \theta^{-1}$ in the limit of small contact angles. As a result, $\dot{R} \propto R^{-9}$ and, finally, we obtain Tanner's law:

$$R \propto t^{\frac{1}{10}}. \qquad (8)$$

Here, we emphasize that the exact integration of the differential equation assumes initial conditions (radius $R_0$ at time zero) so that $R^{10} - R_0^{10} \propto t$. Hence, a simple testing of an exponent from a $R(t)$ log-log plot is only valid if $R \gg R_0$, a condition that is not always
fullfilled in experiments. The validity of Tanner's law also requires volume conservation, no gravity, and that the hypotheses underlying the Cox-Voinov law hold. Hence, another power law is obtained when the radius of the droplet becomes larger than $l_c$. In this case, viscous dissipation balances gravity, and the exponent is 1/8.¹² These power-laws are valid only some time after deposition of a droplet on a substrate,¹³ when dissipation near the contact line dominates the flow. Moreover, while these laws are established under the assumption of complete wetting, $\theta_{eq} = 0$, they may remain valid provided $\theta \gg \theta_{eq}$. Finally, the exponent differs if the liquid spreads in 2D rather than 3D¹⁴ due to a modified volume conservation equation. These remarks support the idea that care should be taken when interpreting droplet spreading data in terms of power-laws to make sure that the underlying assumptions are valid in the systems of interest, and that initial and final conditions are correctly accounted for. Other effects, such as inertia, lead to power laws with exponents larger than 1/10.¹⁵,¹⁶

The simplest test of the Cox-Voinov law consists in the deposition of a droplet on a solid substrate and the measure of the contact angle as a function of the velocity of the contact line during its spontaneous spreading. However, the range of contact angles and velocities is often small. This issue is circumvented by imposing the relative velocity of the contact line with respect to the substrate. Some experimental situations are illustrated in figure 3. Whichever way is chosen to set the contact line in motion, the dynamics at the contact line is governed by the same capillary force as in spontaneous wetting conditions, and the Cox-Voinov law is expected to be valid provided the flow in the drop satisfies the different conditions detailed in the preceding section.

Experimental checks of the Cox-Voinov law are shown in figure 4. The figure is reproduced from the review by Bonn et al.⁶ and compiles data obtained by different authors for oils spreading on surfaces in total wetting conditions. The dynamic angle is much larger than the equilibrium contact angle at capillary numbers much smaller than unity. The experimental data shows an excellent agreement with the Cox-Voinov law. The corresponding value
<figure><img src="image_4.png" /><figcaption>Figure 3: Experimental situations used to measure the dependence of the dynamic contact angle on contact line velocity: (a) the droplet is inflated or deflated, (b) the substrate is moved at velocity *U* while the droplet is maintained immobile, (c) the substrate is tilted and the droplet is set into motion by gravity. For the sake of simplicity, we assume that the front and rear angles are identical, i.e. no contact angle hysteresis.</figcaption></figure>

of the length ratio $x/\ell$ is $10^4$. In the experiments with zero equilibrium angles corresponding to the data of figure 4, a precursor film forms ahead of the line; the thickness of the film is then the natural cut-off length $\ell$. Its value ranges from 1 to 100 nm, which is consistent with both measurements conducted at a distance $x$ from the contact line ranging from 100 µm to 1 mm and a lengths ratio $x/\ell$ of $10^4$. More details about the microscopic structure of the flow near the contact line can be found in the review by Bonn et al.⁶.

In what follows, we focus on effects that modify either the dominant dissipation process or the driving force. The characteristic length scale introduced by those effects may be of the same order as the microscopic cut-off length, as when line friction is at stake, or much larger than the cut off length, when either dissipation in the substrate, Marangoni effects or finite-size effects dominate the dynamics.

## 3 Line friction and retardation mechanisms

An alternate dissipative mechanism to viscous friction may arise from thermally activated displacements of the contact line, either at a molecular¹⁷ or a mesoscopic scale.¹⁸ The primary motivation for the development of the molecular scale theoretical framework, known as molecular kinetics theory, was to solve the apparent contradiction between a zero-velocity
<figure><img src="image_5.png" /><figcaption>Figure 4: Contact angle as a function of the capillary numbers for different experiments in total wetting conditions. The full line represents equation 6 with $\frac{x}{l} = 10^4$. Reproduced from Bonn *et al.*<sup>6</sup></figcaption></figure>

boundary condition at the solid interface and a moving contact line. At the solid interface, as depicted in Fig. 5, molecular adsorption of the advancing liquid and molecular desorption of the receding phase are described within an Eyring framework: molecular jumps between sites are thermally activated and the energy landscape between adsorption states exhibits activation barriers $E^*$ that are biased by the driving capillary force $\gamma(\cos \theta_e - \cos \theta_m)$ where $\theta_m$ is the microscopic contact angle. The process is characterized by the mean distance $\lambda$ between adsorption sites that is expected to be smaller than 1 nm. The jump frequency $K^0$ is set by the biased activation energy. This molecular mechanism should therefore depend on surface chemistry, i.e. molecules adsorbed at the solid surface, or in other words, on the preparation of the surfaces.

An alternate description was offered where the elementary displacements of the contact line occur at mesososcopic scale<sup>19</sup> and are set by the depinning of the contact line from defects or heterogeneities of the solid surface. This picture reconciliates the description of contact line dynamics with the existence of a static contact angle hysteresis<sup>18</sup> in the particular case of non-viscous liquids. In this framework $\lambda$ is a correlation length of the disorder,
and, as such, may be larger than the nanometer size, and $E^*$ is the energy barrier between two pinned configurations of the triple line on defects. As for the molecular mechanism, the energy barrier sets the attempt frequency of the jumps $K^0$.

<figure><img src="image_6.png" /><figcaption>Figure 5: Liquid droplet spreading: Molecular kinetics model. Schematic representations of molecular jumps between sites distant by $\lambda$ with attempt frequency $K^0$. Adapted from Goossens et al.²⁰.</figcaption></figure>

As carefully summarized in the review by Bonn et al.,⁶ the rate of dissipation by the processes acting at the contact line sets the microscopic contact angle $\theta_m$ in the immediate vicinity of the line. This rate is often recast into a friction term, $\zeta U$, where $\zeta$ is a friction coefficient in Pa.s, so that:

$$ \gamma (\cos \theta_e - \cos \theta_m) = \zeta U \quad (9) $$

In this view, the friction coefficient accounts for all localized dissipative mechanisms. In the activated kinetics view, $\zeta$ is related to the elementary jumps length $\lambda$ and frequency $K^0$ through $\zeta = \frac{kT}{\lambda^3 K^0}$. A combination of hydrodynamic and molecular effects has been proposed,²¹ where viscous dissipation sets the relationship between the macroscopic angle $\theta$ and the capillary number through Eq. 6 but in which friction at the line simply appears as a correction to the microscopic contact angle measured at vanishing speed and is set by Eq. 9. For liquids spreading in air, we obtain:

$$ \theta^3 - \left[ \arccos \left( \cos \theta_e + \frac{\zeta U}{\gamma} \right) \right]^3 = 9Ca \ln \frac{x}{\lambda} \quad (10) $$
This combined model can be conveniently extended to liquid/liquid/solid systems using Eq. 7 instead of Eq. 6. Alternate combined models have been proposed where dissipative mechanisms are additive.²²

Very recently, rather than analytical expressions, Blake et al.²³ derived an iterative procedure to treat the angle-velocity data with a model combining the two dissipative mechanisms.²³

Stepping out of the combined model, it is worth emphasizing that the limits within which activated dynamics model and the hydrodynamics model - referred to as Cox-Voinov in the present review - apply are still debated. A transition between hydrodynamic dissipation and contact line friction is predicted by some theoretical works²⁴ as the contact angle decreases during spreading, but this view is sometimes experimentally validated¹¹,²⁵⁻²⁷ or challenged.²⁸,²⁹ When observed, the transition is measured to be set by a critical capillary number, as experimentally measured¹¹,²⁵⁻²⁷ although no consensus exist on its value. It is very likely that the crossover will depend on the ratio between the friction coefficient ζ and the viscosity, as pointed out by De Ruijter,²² rather than by a definite value of the capillary number that would be valid for all systems.

Other experiments evidenced friction at the contact line, as defined by Eq. 9, but it was not always ascribed to a thermally activated process: some studies relate instead the phenomenological friction coefficient ζ to large-scale (micrometer) surface heterogeneities at the solid surface.²⁹,³⁰ In experimental systems where great care was taken to gain control on the state of the solid surface (chemical homogeneity and impurities; partial, pseudo-partial or total wetting regime) with well-designed topological defects (spherical caps with typical width 100 nm and height ranging from 10 to 100 nm), and controlled contact angle hysteresis using polymer brushes adsorbed at the solid interface, the authors conclude that there is no evidence of thermal activation specifically due to added defects.³¹ Instead, Lhermerout and Davitt³¹ offer that energy dissipation takes place in the swollen polymer brush, a mechanism that will be further discussed in Section 4, and that would introduce another length scale
setting the activation length. Finally, the authors point to the need of new experimental methods allowing to discriminate between chemical and topographical heterogeneities, or address collective effects in contact line pinning.³²

Whether it involves contact-line friction or hydrodynamic dissipation, contact line dynamics appears to involve a micro- or mesoscopic scale that either sets the activation length or the flow boundary condition. In this view, it is crucial to gain a good control on the physico-chemistry of each of the three phases : molecular or topographical heterogeneities of the substrate, slip length or molecular ordering at solid interfaces, prewetting films ahead of the line, evaporation, condensation or surface/bulk diffusion of the liquid(s). Below, the relevance of the precise control of the physico-chemistry of the experimental systems in terms of activated dynamics will be further discussed along three lines : prewetting films, heterogeneities and transfers of surface-adsorbed molecules, and, finally, electrostatic surface charges.

## 3.1 Molecular adsorption mechanisms and prewetting films

To the best of our knowledge, the possible presence of precursor films is rarely acknowledged in studies reporting measurements of friction at the contact line.²⁷ Few studies couple the characterization of wetting dynamics to observations of the structure of the liquid close to the contact line at the nanometer scale.³⁵ However systems such as high energy surfaces (clean glass, silica or native oxide layer of silicon wafers in air...) are likely to exhibit total wetting conditions and to give way to the existence of a prewetting precursor film. In such a case, molecular or depinning jumps are expected to be drastically modified. An example is shown in Fig. 6. For droplets consisting of polarisable polymer melts spreading on silicon wafers, observations of the wetting dynamics at the macroscopic scale proves the existence of an equilibrium contact angle³³ while measurements at the microscale reveals the presence of a nanometer-thick precursor film.³⁴,³⁶ This system is a case of the so-called pseudo-partial
<figure><img src="image_7.png" /><figcaption>Figure 6: (a) Pseudo-partial wetting of polyisoprene (PI) melts on oxidized silicon wafers with varied terminal end-groups. (b) Dynamic contact angle decreases over time towards a finite non zero contact angle. Data can be fitted to a friction model as in Eq. 9. From Liu et al.³³. Depending on the end groups, both equilibrium contact angle and friction ζ vary. (c,d) Ellipsometric microscopy images from Schune *et al.* showing a prewetting precursor film of nanometer thickness *h* spreading (c) around a small droplet of PI-OH over time³⁴ and (d) a the edge of a large droplet of PI-CH3. The equilibrium contact angle, the dynamics and the precursor film thickness (or molecular surface density) all strongly depend on the terminal groups. Note the thickness scale change between (c) and (d).</figcaption></figure>
wetting state$^{37,38}$ that can be detected only if observations at the nanoscale are performed.

The analysis of both the dynamics and the statics should integrate this peculiar boundary
condition.

<figure><img src="image_8.png" /><figcaption>Figure 7: Precursor films : Schematic representations of molecular jumps between sites distant by *a* with probabilities *p*, *q* to model the thermally-activated spreading of molecular thin films on a solid wall connected to a reservoir. Reproduced from Burlatsky et al. $^{39}$.</figcaption></figure>

Similar molecular dynamics arguments as those invoked in molecular kinetics theory,
depicted in Fig.5, appear in the modelling of both wetting dynamics and precursor films
spreading,³⁹,⁴⁰ as depicted in Fig. 7. Measurements of the activation energies of molecular
jumps at the solid surface, obtained from the spreading dynamics of the molecularly-thin
films that consisted of quasi-2D gases, validate the latter description.³⁶ These observations
demonstrate that thermally activated motion of molecules at the contact line are very effec-
tive at pulling a film of molecular thickness out of the drop (Fig. 7) rather than setting the
whole liquid corner in motion. We note that experimental works have shown that prewetting
films evolves from 2D-gases at early stages³⁶ to dense thicker films at later stages.³⁴ This
certainly has consequences on a molecular kinetics interpretation of the spontaneous spread-
ing of droplets : in the former case, free adsorption sites are available while they are not
in the latter, meaning that both the length λ and the frequency $K^{-0}$ of the jumps could be
time-dependent. The transition from dilute to dense prewetting films also has consequences
for the viscous hydrodynamics model, although to a lesser extent : the cutoff length $\ell$ in
Eq. 6 can be taken as the film thickness and thus increases over time from 1 nm up to 10 nm.
However, the dependence of the $\theta_d(V)$ law on this cutoff length is logarithmic, and hence rather weak. Finally, whatever the dissipative mechanism, the *equilibrium* contact angle is expected to depend also on the solid/air interfacial tension $\gamma_{sv}$ which of course depends on the coverage of the solid by the precursor film, as theoretically described by Brochard-Wyart et al.³⁷. A possible consequence of prewetting films is the striking behaviour of polymer melts known as autophobic wetting:⁴¹,⁴² it is observed that the contact line recedes after the initial spreading stage of a melt droplet. This effect may result from the time-dependent adsorption of polymers on the solid within the prewetting film.

Altogether, these remarks call for new experimental data combining observations at all scales of the contact line dynamics on well-controlled systems, which are made easier by the recent development of experimental techniques.³¹,³⁶,⁴³

## 3.2 Desorption of physisorbed molecules from the solid by moving contact lines

<figure><img src="image_9.png" /><figcaption>Figure 8: Schematic illustration of a contact line advancing on a solid on which adsorbed molecules form heterogenous patches. (i) If adsorption energies are weak enough, the line desorbs some of the molecules. (ii) The chemical heterogeneities pin the line. Reproduced from Franiatte et al.⁴⁴</figcaption></figure>

As pointed out earlier, we expect molecular dynamics at the contact line to depend not only on the nature of molecules adsorbed at the solid interface, but also on the length scales characterizing their adsorption as well as the anchoring energy of the line on defects. In the
following, we review experimental examples where an additional effect comes into play: the removal of weakly adsorbed species by the contact line itself (Fig. 8). Indeed, the capillary energy associated with a contact line is typically of the order of a few $kT$ so that any molecule or particle desorption process involving activation energies smaller than capillary energies of a few $kT$ is likely to occur. For example, it was recently evidenced that a capillary force at a moving contact line can displace particles,⁴⁵ and, in the present review, extraction of mobile free chains from elastomers by contact lines will be reported in Section 4. The two examples we review below demonstrate that the knowledge of the solid surface state at a molecular level is necessary to compare wetting dynamics data with models, but is not sufficient if the moving contact line modifies the surface.

A recent experimental study of the wetting dynamics of oil droplets immersed in water and spreading on silica⁴⁶ evidences that cationic surfactants adsorbing to silica considerably slow down spreading. To do so, an oil droplet is pressed onto a silica surface at equilibrium with a surrounding solution containing CTAB at varied concentration. By imaging the wetting of oil using a combination of interferometric microscopy and macro imaging, Rondepierre et al.⁴⁶ probed the wetting dynamics over large range of contact angles and velocities (Fig. 9a). They showed that both the static and dynamic wetting data depend on the surface concentration of surfactants at the water/silica interface $\Gamma$ rather than their bulk concentration. By gently measuring molecular adsorption morphologies (Fig. 9b) with Atomic Force Microscopy (PFT-AFM) Kekicheff and Contal⁴⁷ show that surfactant patches with increasing characteristic length grow at the surface of the solid as their bulk concentration increases up to a threshold $\Gamma^*$. In this lower range of concentration, the spreading velocity of oil was measured to decrease by several orders of magnitude with increasing surfactant concentration, as reported in Fig. 9a); meanwhile, the statics contact angle showed that the solid was more and more hydrophobic. This apparent contradiction is overcome by showing that the dissipation mechanism at play is a friction localized at the contact line: an
<figure><img src="image_10.png" /><figcaption>Figure 9: (a) Wetting dynamics of oil on silica in water with CTAB cationic surfactants at concentrations 0, 0.15 and 0.7 times critical micellar concentration from cyan to dark blue.⁴⁶ Contact angle versus velocity curves are fitted to the friction model Eq. 9 (lines) with ζ and θₑ as fitting parameters (see d,e). The Cox-Voinov model Eq. 7 (dotted line) fails to fit the data. (b) Morphology of CTAB-laden mica surfaces carefully imaged in water by PFT-AFM by Kekicheff and Contal⁴⁷. As bulk concentration in CTAB increases, surface concentration Γ increases. First, patches of electrostatically adsorbed surfactants grow until Γ* where a bilayer starts to form through weaker hydrophobic or entropic interactions. (c) Γ* also separates two wetting regimes : at low Γ, patches of adsorbed surfactants are not modified by the moving contact line, and increase both dissipation ζ and hydrophoby S; at larger Γ, surfactants from the top of the bilayer are desorbed by the moving contact line and no longer contribute to friction or hydrophoby. (d) Contact line friction ζ as a function of surface density of surfactants and (e) Affinity of oil for surfactant-laden silica measured as S = γ(cos θₑ − 1).⁴⁶</figcaption></figure>
equation similar to Eq. 9 adapted for two fluids, was found to best fit the velocity-contact
angle data introducing a line friction coefficient $\zeta$. The friction coefficient $\zeta$ increases with
the surface concentration of surfactant $\Gamma$ at the solid/water interface (Fig. 9d) below $\Gamma^*$ as
a consequence of the increased heterogeneity of the silica surface induced by the presence
of the hydrophobic surfactant patches. In this low-concentration regime, surfactant adsorp-
tion is strong enough that imaging the surface at rest provides a good enough picture of
the physico-chemistry of the interface at molecular scales to draw a link with friction and
hydrophoby. Indeed, at low concentration, adsorption involves electrostatic interactions that
are usually stronger than capillary energies, of order 1 to 10 $kT$. Hence, adsorbed surfactants
are unlikely to desorb under the action of the moving contact line.

However, weakly adsorbed surfactants may desorb, and this is what is observed above the
threshold concentration $\Gamma^*$, where the first monolayer is now complete. Above $\Gamma^*$ additional
surfactants start to form a double layer and adsorption then proceeds through weak entropic
(or hydrophobic) interactions. Conversely, both the spreading parameter $S = \gamma (\cos \theta_e - 1)$
of oil and friction coefficient $\zeta$ no longer depend on the surfactant concentration. They
stick to the value set by the first strongly adsorbed monolayer underneath. Hence the
oil/water/solid contact line is found here to be able to desorb weakly adsorbed molecules.

The desorption of molecules from a solid surface by moving contact lines is also evidenced
in an air/liquid/solid configuration where the solid exposed to the atmosphere is contami-
nated by traces. To do so, Franiatte et al.⁴³,⁴⁴ use AFM tips shaped as needles of radius
$60 < r < 300$ nm that are dipped into and retrieved from a droplet at constant velocity $U$.
By measuring the force exerted on the tip by the contact line, the authors first demonstrate
that contact line motion displays discontinuities that reveal topographical defects on the tip
surface. These topographical defects are characterized by a nanometer-sized lengthscale that
is used in the modelling of the dynamics of the whole contact line: the velocity dependence of
the capillary force obeys an activated dynamics model as in Eq. 9. One major contribution
of this work is the discrimination between the effects of topography and the chemistry of the
defects. Indeed, the authors show that the nature and number density of molecules physically
adsorbed on the silicon oxide layer of the AFM tip set the amplitude of the activation barrier
$E^*$ in the kinetics model while topography sets the lengthscale $\lambda$. One other major finding
is that physisorbed molecules, probably airborne short-chain hydrocarbons always present
in atmosphere, are efficiently removed after a few tens of dipping-retrieving cycles. As a
consequence, both the statics and the dynamics of the contact line are modified, through $\theta_e$
and $\zeta$ respectively. This effect introduces another velocity-dependence: indeed, the desorp-
tion by the contact line shifts the adsorption equilibrium between contaminants in air and
at the solid surface all the more as the line moves faster and faster. Hence, the apparent $\theta_e$
and $\zeta$ measured in ambient atmosphere could vary with velocity and the number of times
the line has swept the surface. The authors conclude that desorption of weakly adsorbed
contaminants by contact lines could explain the variability often observed in contact angle
measurements, especially with polar liquids and high energy surfaces.

Finally, it is worth underlining here the similarity between the mechanisms involved in the
capillary-driven desorption of molecules and the Schallamack model<sup>48</sup> for rubber friction
where polymer chains are adsorbed across the solid interface, then stretched upon sliding
and dissipate energy as they detach. In both cases, the elastic energy stored in the defor-
mation of the contact line in contact line friction and in the stretched polymer chains in
elastomer friction is spent in a molecular desorption event : desorption of contaminants in
the former case, and of the polymer chain across the moving interface in the latter case. The
elastic energy associated with both processes can be estimated<sup>49,50</sup> and is found to be of the
order of $10kT$ for most liquids and rubbers, which is enough to break physical bonds but not
chemical or covalent ones (of the order of 100 kT).
<figure><img src="image_11.png" /><figcaption>Figure 10: Electrostatic retardation of water drops (33 µm) sliding down a series inclined plates (50°) of varied material and surface treatment. The drop is initially grounded. Images are taken every 10 ms from the side. While silicon (Si) or gold wafers show no electrostatic charge of the drop, silica (SiO₂) plates of low permittivity charge the drop and slow it down. Both the charge and the friction force depend on the slide distance, the substrate thickness, and on the drop number. (g) Schematics of mechanisms involved in the charge transfers and model developed to relate the electrostatic friction force to the interfacial charges deposited at the air-solid interface and to the drop charge. Reproduced from Li et al.⁵¹.</figcaption></figure>
### 3.3 Electrostatic retardation

Very recently, the motion of sliding droplets of water or other polar liquids on low-permittivity substrates was measured to be substantially influenced by electrostatic forces⁵¹,⁵² which amount to an extra friction term that can make up to half of the total dissipation measured. The mechanism underlying this electrostatic retardation is ascribed to the deposition of electric surface charges of the substrate behind the sliding drops. For water on hydrophobic surfaces, the drops usually acquire a positive charge while interfacial charges are negative, their origin still being debated. By setting up carefully designed experiments where droplets slide down an inclined plate (Fig. 10a), Li et al.⁵¹ are able to measure the electrostatic retardation force by comparing high and low-permittivity substrates having the same surface treatment. The droplet charges are also measured. These inputs feed a model accounting for the Coulomb force exerted on the charged droplet by the electric field created by the surface charges on top of the dielectric substrate; this model successfully describes droplet motion and the electrostatic friction force⁵² (Fig 10g). We believe that these findings shed new light on existing experimental data: indeed, the friction force exhibits several features that have been overlooked so far, such as its dependence on history through the charges left from previous drops (Fig. 10c,d), its non-monotonic dependence on slide length and on the thickness and the permittivity of the underlying substrate (Fig. 10b,c). Furthermore, the experimental systems used by Li et al.⁵¹ are ubiquitous (among others, water, NaCl+water, ethylene glycol as liquids; fluoro-silanated glass or silicon wafer for solids) and found in numerous wetting dynamics studies.²⁸ We also note that the neutralization of charges at the solid air interface is expected to depend on the ambient humidity, which should then be controlled in dynamic wetting experiments. Finally, Stetten et al.⁵² suggest that electrostatic effects on wetting dynamics could extend to hydrophilic and conducting surfaces as well: shorter relaxation times for charge neutralization are expected in these cases and the effect may have remained unnoticed. Yet, even short-term charging may influence wetting dynamics. Experimental data collected on conductive and high permittivity substrates showed however that electro-
statics is negligible:⁵¹ the difference between gold and silicon seen in Fig. 10a,e,f) reduces to a static contact angle effect. The good news is then that past experiments performed on silicon wafers that are semi-conductors were probably not affected by electrostatics.

We believe these findings on electrostatic retardation of wetting dynamics open a wide field for experimental research. The coupling between charge transfers and contact line movement evidenced above could introduce a new length scale setting the activation length. Hence, assessing the charge transport at the solid surface as depicted in Fig. 10g-2) could allow for a quantitative modelling of the charge distribution over time and deepen our understanding of the couplings between slide electrification and retardation.

In summary, we have reviewed experiments in which the dynamics of the contact line is coupled with transfers at the molecular scale. The couplings are mediated by chemical adsorption on the solid that results in energy dissipations that are much larger than that by viscous friction, inducing large decreases of the contact line velocity. In the following section, we will also explore alterations to the substrate caused by the moving contact line, but these modifications are of a mechanical nature rather than chemical.

# 4 Dissipation in the substrate

## 4.1 Soft non-soluble substrates: deformation at the contact line

We now examine the case in which energy dissipation takes place in the substrate rather than in the liquid, as in Cox-Voinov model (Section 2), or at the contact line, as in Section 3. The Young-Dupré law, given by Eq. 1, accounts only for force components tangential to the solid surface, the normal force balance being left aside. However, very early, Lester⁵³ shows that the normal component of the resulting capillary forces at the contact line deforms the
solid with a typical amplitude given by

$$h \propto \frac{\gamma}{\mu_0}, \qquad (11)$$

with $\gamma$ the surface tension of the liquid/gas interface and $\mu_0$ the shear modulus of the solid. For solids such as glass, with shear moduli greater than 1 GPa, this deformation is subatomic, and thus negligible. If the shear modulus is rather that of a very soft solid, $\mu_0 \simeq 10$ kPa, and the typical value of the surface tension of the liquid, $\gamma \simeq 40$ mN m$^{-1}$, we obtain $h \sim 4$ µm.

The resulting ridge formed at the contact line has important consequences on the dynamics of wetting. For example, droplets moving on soft layers covering the surface of a rigid solid slow down compared to the case of the bare surface.⁵⁴⁻⁵⁷ Since contact line motion imposes that the ridge propagates along the surface of the viscoelastic substrate, energy dissipation in both the liquid and the solid occurs. As a consequence, the velocity of the contact line for a given difference between the dynamic contact angle $\theta$ and its equilibrium value $\theta_e$ is smaller on a viscoelastic substrate that acts like a viscoelastic brake.⁵⁷

In most experiments, the substrates are made of cross-linked silicone rubbers. Their shear modulus can easily be tuned by changing the cross-linker concentration. They are often referred to as silicone gels, although they do not contain any solvent. We will see later that they contain free (uncross-linked) and tethered chains but in a first approach we neglect the effect of these chains. The rheology of rubbers is characterized by a single relaxation timescale $\tau$. Dimensional analysis of the problem shows that the dynamics of moving contact lines depends on the relative magnitude of energy dissipation in the liquid and the solid⁵⁸,⁵⁹ estimated with the relaxation ratio,

$$\mathcal{R} = \frac{\gamma\tau}{\eta\ell_s}, \qquad (12)$$
where η is the viscosity of the liquid and ℓ_s is the elastocapillary length of the solid,

$$ \ell_s = \frac{\gamma_s}{2\mu_0}, \qquad (13) $$

with γ_s the surface tension of the gel taken as the average between its interfacial tension with the liquid γ_sl and its surface tension with the atmosphere γ_sv. This length scale is the characteristic amplitude of deformations, since Eq. 11 can be rewritten as $h \propto (\gamma_{lv}/\gamma_s)\ell_s$. The relaxation ratio is the proportionality factor between the liquid capillary number, Ca, and its solid counterpart, $Ca_s = RCa = U\tau/\ell_s$.

To the best of our knowledge, most studies in the literature are performed in the regime corresponding to $R \gg 1$, i.e. when dissipation mainly occurs in the solid substrate, and the viscosity of the liquid does not affect droplet motion.⁵⁸ Careful measurements of the dependence of the dynamic contact angle θ on the velocity U of the contact line show that they still relate to each other through a power law at small values of the capillary number, with an exponent that differs from that expected in the Cox-Voinov law. Scaling arguments similar to those leading to the Cox-Voinov relation can be used to explain this relation. The capillary power per unit length of the contact line injected in the system is

$$ P_{in} \sim \gamma U (\cos \theta_e - \cos \theta). \qquad (14) $$

The ridge has a typical cross-section $\ell_s^2$. At frequency ω, the dissipated power per unit length in the ridge scales as

$$ P_d \sim \mu'' \omega \epsilon^2 \ell_s^2, \qquad (15) $$

where ϵ is the scale for strain and μ'' the loss modulus of the substrates. The loss modulus of the silicone gels used in experiments is a power law of frequency, μ'' = μ_0(ωτ)^m. Taking ϵ = γ/γ_s sin θ as a scale for strain and ω = U/ℓ_s as the typical frequency at which the contact
line deforms the solid, we obtain

$$ \mathcal{P}_d \sim \mu_0 \left( \frac{U\tau}{\ell_s} \right)^m U \ell_s \left( \frac{\gamma}{\gamma_s} \sin \theta \right)^2. \qquad (16) $$

Equating Eq. 14 and 16, the following relation is finally found

$$ f(\theta) \propto \frac{\gamma}{\gamma_s^2} \mu_0 \ell_s (R C a)^m, \qquad (17) $$

with

$$ f(\theta) = \frac{|\cos \theta - \cos \theta_{eq}|}{\sin^2 \theta_d}. \qquad (18) $$

The absolute value allows comparison of data obtained for advancing and receding contact lines.

Figure 11 shows that this scaling law is in good agreement with available experimental data$^{60–63}$ for the dynamic contact angle $\theta$ at small contact line velocities over silicone substrates with different properties. Note that this figure shows data for which we were able to retrieve all the parameters to compute $\mathcal{R}$. Other data follow the general dimensionalized law $f(\theta) \propto Ca^m$.64

The experimental data deviate from Eq. 17 at the largest values of the solid capillary number. The slope decreases as $Ca_s$ increases. In this regime, the characteristic timescale of the flow is smaller than the relaxation timescale of the substrate. As a consequence, the storage and loss modulii of the gel are of the same order of magnitude and both contribute to its apparent stiffness that increases with frequency. The size of the ridge is constrained, and so is dissipation. Recent insights obtained from a non-linear description of the problem$^{59}$ suggest that the curve tends toward an asymptotic value,

$$ f(\theta) \underset{Ca_s \to \infty}{\longrightarrow} \frac{\gamma m}{\gamma_s (1-m)}. \qquad (19) $$

This equation still requires extensive testing as only a limited set of data for the saturation
<figure><img src="image_12.png" /><figcaption>Figure 11: Summary of available contact angle data sets plotted as a function of the *m*-th power of the solid capillary number *Ca*<sub>s</sub>, where *m* is the exponent appearing in the loss modulus of the substrate. Properties of the substrate: ○ shear modulus μ<sub>0</sub> = 1200 Pa, relaxation time τ = 130 ms, loss exponent *m* = 0.55;<sup>60</sup> △ μ<sub>0</sub> = 1.08 kPa, τ = 15.4 ms and *m* = 0.626;<sup>61</sup> □ μ<sub>0</sub> = 265 Pa, τ = 480 ms and *m* = 0.61;<sup>62</sup> ∇ μ<sub>0</sub> = 390 Pa, τ = 540 ms and *m* = 0.58 and □ μ<sub>0</sub> = 415 Pa, τ = 80 ms and *m* = 0.61.<sup>63</sup> We were unable to find the value of the equilibrium contact angle of the liquid on the PVS sample. As this material is similar to PDMS, we assume that θ<sub>eq</sub> = 105 °. Dot-dashed line: scaling given by Eq. 17.</figcaption></figure>
exists. Finally, the contact line enters a stick-slip regime$^{60,62,65}$ in which it detaches from the tip of the ridge and moves fast until another ridge has had the time to grow.

Energy dissipation in the substrate may be an advantage or a drawback in applications. Several system parameters provide control over dissipation. For example, when the substrate is a layer of rubber coating a rigid substrate, Zhao et al. 61 report a decrease of dissipation as the thickness $h_0$ of the rubber layer decreases. They interpret this observation as a finite-depth effect that results in the decrease of the ridge height as the layer is thinner. This effect becomes prominent when $h_0 \le \ell_s$. They suggest that, in the latter regime, the relevant length scale for the height of the ridge is

$$h = \left( \frac{\gamma_s h_0^3}{\mu_0} \right)^{1/4} \qquad (20)$$

rather than $\ell_s$. A scaling model describing the motion of the contact line for thicknesses of the order of the elastocapillary length well captures the decrease of dissipation. More recent work by Khattak et al. 66 shows that scaling 20 fails when $h_0 \ll \ell_s$. Indeed, the strain $h/h_0$ diverges as the thickness decreases, and the deformation cannot be described in the frame of linear viscoelasticity. Khattak et al. 66 suggest a scaling based on the assumption that the height of the ridge is proportional to $h_0$, which seems to capture the trend of the experimental data. Another way to tune the amplitude of the ridge is to stretch the soft layer. Snoeijer et al. 67 measure the dynamic contact angle and show that the contact line moves faster on the stretched substrate than on that undeformed for a constant value of the dynamic contact angle. Smith-Mannschott et al. 68 show that droplets slide the fastest down a vertical stretched sheet when the stretch direction is parallel to that of gravity. They measure the surface profile of their stretched layer around the droplet (Fig. 12a). They observe that the amplitude of the ridge is the smallest along a line passing through the centre of the droplet and parallel to the stretch direction. Here too, a smaller ridge results in a smaller dissipated energy.
<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /><figcaption>Figure 12: (a) Profilometry of the surface of a stretched soft layer around a sessile droplet. Applied strain $\epsilon = 60\%$. Taken from Smith-Mannschott et al.⁶⁸. (b) Velocity of a droplet sliding on a textured soft surface and on a smooth soft surface as a function of the inclination of the surface. Reproduced from Coux and Kolinski⁶⁹.</figcaption></figure>

Another path to control dissipation is to texturize the surface of the substrate. Coux and Kolinski⁶⁹ show that the motion of droplets sliding down the surface of a soft silicone gel patterned with pillars is insensitive to viscoelastic braking. They demonstrate that the solid contribution to dissipation is small because the droplet contact line induces significant deformations of the solid only at its trailing-edge contact line, via pillar bending. The associated dissipation is orders of magnitude smaller than that in the liquid. Hence, only the latter contributes to the dynamics of the droplet, and droplets slide on patterned soft surfaces orders of magnitude faster than on smooth ones (Fig. 12b).

The description of the dynamics of wetting on soft viscoelastic solids yielding Eq. 17 date back to the 1990’s⁵⁶ and rely on a linear viscoelastic description of the substrate: the deformations of the gel are assumed infinitesimal. Within this framework, surface slopes on the gel are proportional to the ratio of the surface tensions of the solid and the liquid, $\gamma_s/\gamma$. Hence, the validity of a linear viscoelastic description is only guaranteed when:

$$ \frac{\gamma_s}{\gamma} \ll 1. \qquad (21) $$
This condition is never met in experiments, since the surface energy of the materials commonly used is of the same order of magnitude as or larger than that of the liquid. Although a model based on linear viscoelasticity can capture the general trends observed in experimental data, it falls short when it comes to providing a quantitative description.

Non-linear descriptions accounting for the finite value of $\gamma_s/\gamma$ have been proposed in recent years, based on different hypotheses regarding the conditions at the surface of the solid. One of these models,⁷⁰ which assumes that a stress singularity exists at the tip of the ridge, is able to describe quantitatively the statics of wetting on soft solids without any adjustable parameter. Similar models for the dynamic case have been proposed.⁵⁹,⁷¹ The predictions of Dervaux et al.⁵⁹ have been tested against experiments with success,⁷² in particular by exploring the relation between the relaxation ratio $R$ and the properties of the moving contact line. However, these problems have to be further explored, in particular in light of the properties of the rubbers that are used in wetting experiments. As pointed out above, they generally contain free polymer chains and we emphasise in the following section the consequences of the presence of this liquid, which results in transfers of molecules from the soft solid to the liquid/air interface.

## 4.2 Coupling of dissipation in the substrate with molecular transfers

### 4.2.1 Capillary extraction of free chains from rubbers

The silicone rubbers or "gels" used in all of the wetting experiments we have reviewed so far are easy to prepare, rather insensitive to surface contamination by impurities and their mechanical properties are easy to vary and control. However, their formulation is hardly controlled, in sharp contrast with the care with which elastomers are usually prepared in the adhesion and friction community.⁷³,⁷⁴ In wetting experiments, the mechanical properties of the silicone gels are tuned by changing the mass ratio of the curing agent and silicone
oil. This change in the formulation comes at the expense of an increased amount of mobile chains in the sample. For example, Sylgard™ 184 prepared under recommended conditions contains around 5wt% mobile chains.⁷⁵ A gel prepared with the same kit with 1 volume of curing agent for 60 volumes of base PDMS contains more than 50wt% mobile chains.⁷⁶ Methods to extract mobile chains without damaging soft samples⁷⁷ have been developed only very recently,⁷⁶ hence many wetting experiments of the literature have been conducted in presence of mobile chains.

Recent investigations show that these mobile chains also play a crucial role in the dynamics of wetting on silicone elastomers, as could have been expected from early studies on the adhesion of elastomers: free mobile chains were shown to play a crucial role in their effective interfacial energy.⁷⁸,⁷⁹ Hourlier-Fargette et al.⁷⁵,⁸⁰ show that droplets residing on a stiff Sylgard™ 184 gel can extract part of the small amount, 5wt%, of mobile chains present in the substrate. The typical extraction timescale induced by a moving droplet is much shorter than when the droplet is sessile. This extraction has a direct consequence on droplet motion. The velocity of a droplet sliding on a vertical non-extracted silicone sample changes over time from a small constant value up to a higher one. Hourlier-Fargette et al.⁷⁵ argue that this transition is related to the contamination of the liquid/gas interface of the droplet by mobile chains coming from the substrate. Indeed, a droplet sliding on a “dry” slab of gel from which mobile chains were extracted moves with constant velocity. The authors relate the time at which the transition occurs to the saturation of the liquid/gas interface with mobile silicone chains. As a result, the surface tension $\gamma$ decreases as well the equilibrium contact angle. Images of the droplet before and after the transition show no significant change of the dynamic contact angle, suggesting that the decrease of the surface tension of the droplet leads to an abrupt increase of the capillary energy injected in the system, hence a sharp increase in the velocity.

Mobile chain extraction has since then received significant attention. Wong et al.⁸¹ bring evidence that their extraction induces a two to threefold increase of the contact hysteresis
of PDMS samples. They highlight the lubricating role of mobile chains. An investigation of the hysteretic response of the extracted samples in connection with the known increase of the roughness of PDMS samples after extraction would be interesting.⁸²

Confocal microscopy reveals the extent to which mobile chains contribute directly to the ridge. Dying the cross-linked PDMS network and the silicone oil with two different fluorophores, Cai et al.⁸³ show that mobile chains occupy an increasing fraction of a static ridge as their amount increases in the substrate. A key finding is that the deformation of the cross-linked network is a non-monotonic function of the swelling amount. Hauer et al.⁸⁴ extend these observations to the case of a moving contact line. They show that silicone oil occupies a smaller fraction of the ridge as the contact line moves faster. They attribute this dependence to a competition between diffusion of the mobile chains in the cross-linked network and their advection by the contact line. These results suggest that the lubricating role of the mobile chains identified by Wong et al.⁸¹ may depend on the velocity of the contact line.

These observations imply that the substrate should rather be described within a poroelastic framework in which the liquid inside the gel rearranges when the substrate is under stress. Such a description exists and has been tested to some extent for equilibrium.⁸⁵⁻⁸⁷ To the best of our knowledge, a similar description for dynamic situations does not exist yet. A poroelastic description will also be interesting for solids containing solvents such as hydrogels.

### 4.2.2 Soluble substrates

Transfers associated with a higher dissipation in the substrate than in the liquid also take place in soluble substrates. Indeed in this case, in contrast to the preceding section, it is the wetting solvent that is transferred to the substrate. When a drop of solvent spreads over a polymer substrate, both its contact angle and velocity are reported to decrease with time.⁸⁸ The angle is plotted as a function of velocity in Figure 13 for water spreading on a micron-
thick polysaccharide layer. The corresponding capillary number remains very low: neither
the surface tension nor the viscosity of the water droplet varies significantly over the course
of the experiment, so the capillary number is at most 10<sup>-5</sup>. Consequently, viscous dissipation
is found negligible and the Cox-Voinov law fails at explaining the large angle variations with
velocity that are reported. In fact, the dynamics is governed by the solvation of the substrate
ahead of the contact line, which can be measured from thickness variations as displayed in
Fig. 13. When spreading starts, the polymer layer is weakly hydrated and behaves like
a rigid hydrophobic substrate. The spreading angle decreases as more water is transferred
to the substrate. Towards the end of spreading, the substrate has a high water content
and the contact angle tends towards zero. Since the hydration of the substrate depends
on the velocity of the drop, the angle also depends on the velocity. More precisely, in this
picture, the contact angle coincides with the equilibrium angle θ ≃ θ<sub>e</sub>. The equilibrium
angle itself depends on the volume fraction of solvent in the substrate, θ<sub>e</sub> = θ<sub>e</sub>(φ) since
the surface energies of the liquid/substrate and air/substrate interfaces should depend on
this. The solvent content varies with drop velocity, φ = φ(U) and, finally, the angle varies
with velocity, θ = θ(U), although viscous dissipation is negligible. Note that the amplitude
of angle variations are much larger for polar than nonpolar solvents. In the former case,
the polymer layer behaves as a solvophobic substrate at the early stages of spreading since
polymer chains at the interface with air tend to exhibit their more nonpolar parts. Hence,
changes in the surface tension of the solvated solid were shown to be set by characteristic
times that depend on the polymer molar mass : shorter chains reorient their polar groups
faster than longer ones.<sup>89</sup> This effect does not exist with nonpolar solvents for which angle
variations were found to be smaller.<sup>90</sup>

If the polymer substrate is initially in a glassy state, large variations in solvent content can
trigger a glass transition during spreading⁹¹ through a plasticization effect. The substrate
becomes highly viscoelastic in the glass transition region and is then significantly deformed
by the contact line. Dissipation occurs in the substrate as the contact line moves, the onset of
<figure><img src="image_15.png" /></figure>

Figure 13: (left) Top view of a droplet spreading on a thin soluble substrate. The colors ahead of the contact line reflect the increased thickness and thus the hydration (color scale) of the substrate. (right) Contact angle as a function of velocity of a drop of water spreading onto a soluble polysaccharide layer. The polymer is initially glassy and glass transition occurs in the course of spreading, when the velocity reaches a value $U_g$ corresponding to an angle $\theta_g$. The dynamics is then partially described following Eq. 17 with $m = 0.75$ (hatched black line). Cox-Voinov law is shown for an equilibrium angle arbitrarily chosen of 5° (grey full line). Adapted from Dupas et al.⁹¹.
which is revealed by a large slope change in the angle vs. velocity curve (Fig. 13): the contact
angle suddenly drops at decreasing velocities (corresponding to increasing time), because en-
ergy is strongly dissipated. Spreading may even stop as the contact line is pinned by the
ridge, as observed with drops of toluene spreading on polystyrene.⁹⁰ Although a quantitative
description is difficult because the driving force and mechanical properties keep varying with
time, the variations of the angle can be locally described by accounting for dissipation in
the substrate in the same way as for nonsoluble substrates, following Eq. 17. Accordingly,
the velocity $U_g$ corresponds to a viscosity of the substrate in the melt phase, just above the
glass transition. In summary, the spreading of solvent drops on glassy soluble substrates can
be strongly modified by the occurrence of a glass transition in solvent content by turning
the substrate into a soft, highly dissipative medium. Away from the glass transition region,
the contact line modifies the properties of the substrate as it moves, which in turn rules
the wetting dynamics. Wetting of soluble substrates therefore constitutes a case of reactive
spreading.⁹²

In the Sections 3 and 4 of this review, we have considered effects on wetting dynamics
arising - at least partly - from the properties of the wetted substrate, such as roughness, ad-
sorbed species, induced charges or viscoelasticity. In the following two sections, we consider
effects that originate from the composition of the liquid.

## 5 Surfactant-induced Marangoni effect

Marangoni effects are negligible when a drop spreads on a substrate on which surfactants
have previously adsorbed, as reviewed in Section 3.2. In contrast, these effects can be at
play when a droplet of surfactant solution spreads on an initially bare substrate. Numerous
studies have addressed the wetting behavior of surfactant solutions, with considerable atten-
tion directed towards surfactants that promote water spreading on low-energy hydrophobic
surfaces such as polymeric surfaces. Water droplets are expected to only partially wet these materials, but, in contrast, droplets of aqueous solutions of specific surfactants completely wet these substrates. These surfactants are called superspreaders, and they have been the object of extensive research.⁹³,⁹⁴ In contrast, other surfactant solutions were shown to have wetting behaviours close to the one of pure water,⁹⁵ although small differences in the wetting dynamics were observed,⁹⁶ which could not simply be accounted for by the modification of the surface tension and equilibrium angle.

Obviously, the presence of surfactants can result in Marangoni stresses at the surface of the spreading drop provided concentration gradients are created. A naive picture suggests that the existence of a region depleted in surfactant close to the contact line would result in a surface stress directed towards the contact line, and consequently in a faster flow than in the pure liquid case. In contrast, the shear induced by the Marangoni stress is predicted to modify the flow in the drop and to lead to a dissipation larger than in the free-surface flow with the same average velocity.⁹⁷ Marangoni stresses therefore results in a slower wetting dynamics. To the best of our knowledge, this counter-intuitive result has received little attention while it seems consistent with experimental observations. For instance, Tanner's law for droplets of AOT solutions has a prefactor smaller than that of pure water.⁹⁶ The effect remains moderate, in agreement with the weak modification of dissipation that is expected.

To understand the behaviour of superspreaders, adsorption of surfactants at the liq-
uid/solid interface must be accounted for. As the contact line progresses, a flux of surfactants
toward the substrate is established that may be sufficient to maintain a surfactant-free region
at the contact line. The role of adsorption at the solid surface in superspreading has been
emphasised by different authors who developed models,⁹⁸⁻¹⁰⁰ but there is still no unanimity
in the exact mechanism.¹⁰¹,¹⁰² Plausibly, as pictured in Figure 14, the strong Marangoni
stress at the depleted region curves the interface in this region and it is the capillary force
on that curved “clean” interface that drives the wetting, augmented by a contribution of the
Marangoni stress.¹⁰⁰ Therefore, the wetting dynamics is Marangoni-enhanced. In addition, this effect can exist only for a range of surface concentration; the “sink” effect produced by the absorption at the substrate needs actually to be larger than surfactant advection toward the contact line; the flux of surfactant from the interface with air to the substrate must therefore be larger than a minimum value.⁹⁷,⁹⁹,¹⁰⁰ At the same time, the flux must be smaller than a maximum value in order for the depleted zone to be confined in a small region close to the contact line, yielding a large concentration gradient and an efficient Marangoni flow. For a soluble surfactant, the condition on the flux results in a condition on the depletion length $Γ_0/c_0$, where $Γ_0$ and $c_0$ are respectively the surface and bulk concentrations. The latter condition is consistent with the fact that only few surfactants are superspreaders¹⁰⁰ and that adsorption at the solid surface is not sufficient to induce superspreading. Available models need to be further compared to experimental findings and in particular to measurements of the angle variations with velocity.

<figure><img src="image_16.png" /><figcaption>Figure 14: Sketch of the region depleted in surfactant close to the contact line, which is maintained by a flux toward the substrate (left) and on which the resulting Marangoni stress induces a curvature of the contact line (right). Note that adsorbed surfactants are schemed as expected for a hydrophilic surface since the described mechanism does not reduce to superspreaders. Reproduced from¹⁰⁰</figcaption></figure>

In summary, as a drop of surfactant solution spreads, a region depleted in surfactant is created at the contact line. The associated Marangoni shearing modifies the flow in the drop, which tends to increase viscous dissipation and slow down the dynamics (or increase the angle if the velocity is imposed) without significantly altering the shape of the contact line.
In contrast, if the surfactant adsorbs at the solid substrate, the region depleted in surfactant is sustained by the associated “leak” and the large Marangoni stress that results deforms the contact line, which in turn modifies the wetting dynamics and may allow spreading on hydrophobic substrates.

In the next section, we consider liquids containing suspended particles, in which a depleted region also forms close to the contact line. The driving effect is then of steric origin instead of a surface tension gradient.

# 6 Finite-size effects

Complex fluids such as suspensions of rigid particles, emulsions and foams are characterized by a characteristic mesoscopic length, which is the size of the dispersed objects $d$. This size can vary from values typical of the colloidal regime, $d \ll 10^{-6}$ m, up to the millimeter, and can be commensurate with the height of a spreading droplet. It is then natural to wonder how the presence of these objects alter the dynamics of wetting depending on their size. Surprisingly, this question has received little attention. In addition, dispersed systems exhibit complex rheological behaviours and, sometimes, unusual boundary conditions for flows. In this Section, we review examples where finite-size effects in dispersed systems play a role in the wetting dynamics of droplets, either through confinement effects or changes in the flow close to the contact line.

## 6.1 Confinement due to finite-size effects

In this part, we review the coupling between confinement effects and wetting dynamics. Confinement is here defined by the ratio between the height $h$ of the liquid wedge as depicted in Fig. 2 and the typical size characterizing a dispersed phase. As a first example, we explore density-matched suspensions of non-Brownian particles, starting with viscous granular suspensions in the dense regime where the particle volume fraction is large, typically $\phi \ge 20\%$,
and as well as the particle diameter $d_p \ge 10$ µm.

Zhao et al.¹⁰³ study the dependence of the contact angle of a droplet of granular suspensions on the velocity of the contact line. The suspensions they use are Newtonian liquids, with a surface tension identical to that of the suspending liquid.¹⁰⁴,¹⁰⁵ Their viscosity $\eta_s$ is proportional to that of the continuous phase:¹⁰⁶

$$ \eta_s = A\eta_0, \qquad (22) $$

with A an empirical prefactor that depends on the particle volume fraction $\phi$ and its distance to the critical particle volume fraction $\phi_c$ at which the bulk viscosity of the suspension diverges. In contrast, $\eta_s$ does not depend on the particle size.¹⁰⁶

<figure><img src="image_17.png" /><figcaption>Figure 15: Dynamic contact angle of a droplet of granular suspension spreading on a silicon wafer as a function of the suspending liquid capillary number, $Ca_0 = \eta_0U/\gamma$. Markers represent data while dashed lines are fits to the Cox-Voinov relation Eq. 6. Data from Zhao et al.¹⁰³.</figcaption></figure>

As shown in Fig. 15, the cube of the dynamic contact angle is proportional to the capillary number $Ca$ of the experiment, based on the viscosity and surface tension of the suspending liquid. These measurements are therefore consistent with the Cox-Voinov law, given by
Eq. 6 with a zero equilibrium contact angle, in agreement with the very small equilibrium angle of the suspensions on silicon wafers. A shift of the curves to higher values of the angle for identical values of Ca, is also observed, consistently with the viscosity increase due to addition of solid particles. However, this shift is found to be a function of particle size whereas the bulk viscosity of suspensions is not. The apparent viscosity of the suspensions during spreading, ηw, extracted from the experimental shifts is a function of both the particle size dp and their volume fraction φ, decreasing down to the viscosity of the suspending liquid η0 when the particle size is larger than around 150 μm.

<figure><img src="image_18.png" /><figcaption>Figure 16: (a) Top views of the region near the contact line during the spreading of a bidipserse suspension of granular particles. The particles have diameter $d_1 = 10$ and $d_2 = 80$ µm. The fraction of 10-µm particles, $\zeta_{10} = 100 * \phi_{10}/\phi$ where $\phi_{10}$ is the volume fraction of small particles and $\phi$ the total volume fraction, increases from top to bottom. Scale bar: 500 µm. (b) Evolution of the viscosity ratio of granular suspensions $\eta_w = \eta_{s,w}/\eta_0$ where $\eta_{s,w}$ is the apparent viscosity of the suspension when spreading, as a function of the amount of small particles $\zeta_{10}$. Adapted from Pelosse et al.¹⁰⁷</figcaption></figure>

A crucial observation reported by Zhao et al.¹⁰³ is that the region inside the droplet in the vicinity of the contact line is depleted in particles to a degree that depend on their size. With this observation in mind, Pelosse et al.¹⁰⁷ highlight the fact that the Cox-Voinov law
applies to contact angles that are measured at most a capillary length $l_c$ away from the contact line. They revisit a scaling analysis¹⁰⁸ that shows that the height of the liquid/gas interface in this domain ranges between 0 and a typical scale,

$$h^* = (3Ca)^{1/3}l_c, \qquad (23)$$

that is able to capture experimental data of the height of the interface at which the curvature changes sign in the case of a simple fluid. This length scale is a weak function of the capillary number: a tenfold increase of $Ca$ increases $h^*$ by a factor around 2 for a given liquid. It separates the droplet domain where the stress balance involves only capillarity and viscous dissipation from the region where gravity starts to influence the shape of the liquid/gas interface. As $10^{-3} \le Ca \le 10^{-2}$ in the reported experiments,¹⁰³,¹⁰⁷ we have $250 \le h^* \le 600$ µm. These values are of the same order of magnitude as the cut-off around 150 µm found by Zhao et al.¹⁰³. They argue that only particles with sizes much smaller than $h^*$ can affect the measurements of the dynamic contact angles, as they will be able to enter the viscous-capillary region and alter energy dissipation. This conclusion is supported by experiments¹⁰⁷ with bidisperse suspensions that show that the ability of small particles to move around larger ones and reach the visco-capillary region sets dissipation in the corner and in turn dictates the value of the dynamic contact angle (Fig. 16).

The theoretical description of the spreading of granular suspensions on a solid is still incomplete. A closer look at the contact line evidences that the first rows of particles experience strong confinement by both the liquid/gas and solid/liquid interfaces. The height of the interface in this region is of the order of a few particle sizes. As a result, particles arrange in a regular way, forming an ordered region that extends over a few particle diameters. Ordering under strong confinement between rigid walls leads to a decrease of the apparent viscosity of granular suspensions.¹⁰⁹ This effect is also expected to depend on inter-particulate friction and adhesion.¹¹⁰,¹¹¹ This element is absent of current modelling attempts and its contribu-
tion should be investigated in the future. Besides, particles hardly ever leave this region
to go back to the bulk. Particle exchange is unidirectional, from the thicker parts of the
droplet toward the contact line, as more particles can fit along the expanding perimeter of
the spreading droplet. The dynamics of this region are at least qualitatively different from
that of the bulk of the suspension.

Interestingly, the effect on wetting dynamics of particles ordering close to the contact line
has been addressed with Brownian suspensions. For example, nanoparticles with diameters
$d_p \sim 10$ nm promote the spreading of the liquid they are suspended in.<sup>112-114</sup> Spreading is
enhanced because the particles form ordered layers in the vicinity of the contact line and
modify the structural disjoining pressure. These systems are also sensitive to interactions
between the substrate and the particles,<sup>115</sup> the liquid and the particles<sup>116</sup> and to the equi-
librium wetting state. For example, theory predicts that the coupling between nanoparticles
and the precursor film that precedes a droplet of completely wetting fluid should affect the
dynamics and shape of the liquid.<sup>117</sup> To the best of our knowledge, these predictions have
not been tested yet.

Similar finite-size effects as in suspensions are expected in emulsions. The spreading be-
haviour of emulsions has received little attention, although they are the workhorse fluids of
many industries such as cosmetics, food and agriculture, either in manufacturing processes
or final products. Besides, the resort to ink-jet printing in an ever broader range of appli-
cations suggests that this understanding will soon become crucial. Forester et al.<sup>118</sup> show
that the radius of a droplet of water-in-silicone-oil emulsion spreading spontaneously on glass
evolves as a power law of time with exponent 1/10, compatible with Tanner's law, given by
Eq. 8, when the volume fraction of water is low. Spreading is slower for denser emulsions,
and the spreading dynamics deviate more and more from Tanner's law at long times. The
emulsion stops spreading when it contains more than 73% of water. The authors also show
that the region next to the contact line is devoid of dispersed water droplets. Besides, the
contact line exhibits an instability reminiscent of the fingering instability observed when a
surfactant solution spreads on a thin film of pure liquid.¹¹⁹ The continuous phase of the emulsion used by Forester et al.¹¹⁸ is a mixture of silicone oils loaded with surface-active molecules, and, as their liquids completely wet the substrate, it would be interesting to see whether their observations are compatible with a fingering instability of the precursor film.⁶ Mohammad Karim and Kavehpour¹²⁰ study similar systems and they report that Tanner’s law is valid over some range of their experiments. However, coarsening of the water droplets seems to affect their observations and makes it difficult to interpret the dynamics they report. Along the same line, Deblais et al.¹²¹ demonstrate that the wettability of the solid by either the continuous or the dispersed phase must be taken into account in the description of the wetting of solids by emulsions. They also point out the need to account for yield stresses to shear appearing in dispersed systems, a point that is easier to explore in suspensions of soft particles as reviewed in the next section.

## 6.2 Flow dissipation in dispersed systems

Dispersed systems also display a large variety of responses to shear, which are expected to modify the dissipation in the shear flow compared to the simple liquid case described by Eq. 3. In the present Section, we focus on the changes of the shear flow due to finite-size effects in dispersed systems. A comprehensive study of the spreading dynamics of droplets of suspensions of soft particles was conducted in the case of dense suspensions of microgels by Martouzet et al.¹²². In these systems, hydrophilic polymer chains are swollen by water and arranged into a microstructure similar to that of a suspension with a characteristic size $1 \mu m$.¹²³,¹²⁴ As a first result, the microscopic cut-off length that sets the viscous dissipation is found to be equal to the microstructure lengthscale. Then, in the framework depicted in Fig. 2, the flow in the droplet is modified in three ways: a slip boundary condition at the liquid / solid interface, a yield stress for flow, and shear-thinning at large shear rates. We detail below their consequences on droplet spreading. First, on a smooth solid substrate,
a lubrication layer develops between the solid and the first layer of microgels¹²⁴,¹²⁵ and translates into an effective slip length: it favors the droplet spreading at all times. This slip effect, however, disappears on a rough substrate with roughness of typical length equal to the microgel size. The yield stress and the shear-thinning effects operates at different stages of the drop spreading. The yield stress $\sigma_y$, of the order of tens of Pascals, becomes comparable to the shear stress at the later stage of the droplet spreading, and at some point, it stops the spreading before the thermodynamic contact angle $\theta_e$ is reached. Hence, the final contact angle $\theta_f$ depends on the spreading history since liquid was deposited. However, Martouzet et al.¹²² derive a remarkably simple relationship between the final contact angle, the equilibrium contact angle and the ratio between the yield stress and the capillary pressure in the droplet at rest $\gamma/R_f$:

$$ \cos \theta_f = \cos \theta_e + C \frac{\sigma_y}{\gamma/R_f} \quad (24) $$

where $C$ is a constant that depends on the slip boundary condition. Beyond this analytical analysis, a numerical resolution of the full problem allows Martouzet et al.¹²² to describe the measurement semi-quantitatively. Finally, at early stages where the spreading velocity is large, the shear rate is large enough so that the shear stress is larger than the yield stress everywhere in the droplet. The shear-thinning effect sets the wetting dynamics and can be described within the framework developed by Starov⁹⁶,¹²⁶ for power-law fluids. Here, we choose not to review other non-Newtonian effects in droplets spreading and restrict to dispersed systems.

Altogether, and as mentioned in Section 6.1 for non-Brownian granular suspensions, the inter-particulate friction is expected to strongly affect the shear flow, not only close to the contact line where confinement and ordering occurs, but in the whole droplet where the rheological behavior of the suspension may be non-Newtonian. As an exemple, suspensions of frictionless particles often display counter-intuitive mechanical responses such as shear-thickening, while they are good candidate to probe the ordering under strong confinement at
the contact line in comparison to classical frictional$^{103,106,107}$ suspensions. Recent progress
in the synthesis of model frictionless suspensions opens the ability to control interparticle
interactions. The interplay between the confining Laplace pressure and the repulsive inter-
particle interaction may lead to interesting spreading dynamics. It will then be necessary to
track the wetting dynamics both at the droplet scale and close to the contact line, at the
particle scale.

7 Conclusion

In conclusion, the aim of the present article is to review recent experimental works on wetting
dynamics, for which complexity arises from the nature either of the substrate or of the liquid.
The guiding thread we have chosen is a departure of the dynamics from Cox-Voinov law.
In the illustrations of this deviation given above, we have emphasised the key role of the
different emerging length scales, either molecular, microscopic or mesoscopic that may result
in a strong modification of the flow in the drop and/or the dissipation. In addition, transfers
between surface and volume must be accounted for, as they modify the solid surface tension
and thus the driving capillary term. These experimental works all point out the challenges in
properly comparing the wetting dynamics data on complex systems or in complex situations
with a reference experiment in order to separate the effects. As emphasised earlier, several
effects are not fully understood yet, and studies on wetting dynamics have a bright future
ahead of them.

Acknowledgement

The authors thank F. Boulogne, A. Deblais, J. Dervaux and F. Lequeux for discussions during
the preparation of this review. M. R. is grateful to the Research School of Physics at the
Australian National University (ANU) and to Centre National de la Recherche Scientifique
(CNRS) for supporting his stay at the ANU during which this review was written through
a Stjepan Marčelja fellowship and the IEA grant “Coflact” respectively.

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