Soft-lubrication interactions between a rigid sphere and an elastic wall

The fluid-structure interaction between flows and boundaries is a central situation in continuum mechanics, encountered at many length and velocity scales. A classical example is lubrication, where the addition of a liquid film, a lubricant, between two contacting objects, allows for a drastic reduction of the friction between them. Such a process occurs in a large variety of contexts with hard materials, such as roller bearings, pistons and gears in industry (Dowson and Higginson, 2014), or faults (Brodsky and Kanamori, 2001) and landslides (Campbell, 1989) in geological settings. At large velocity, or moderate loading, the liquid film is continuous with no direct contact between the solids. When the solids are deformable, the friction force can be described using elastohydrodynamic (EHD) models within the soft-lubrication approximation (Dowson and Higginson, 2014).

The previous EHD coupling is also widely encountered in soft condensed matter, but at very different pressure and velocity scales (Karan et al., 2018). Examples encompass the remarkable frictional properties of eyelids (Jones et al., 2008) and cartilaginous joints (Mow et al., 1984; Jahn et al., 2016), as well as biomimetic gels (Gong, 2006) and rubbers (Sekimoto and Leibler, 1993; Moyle et al., 2020; Wu et al., 2020; Hui et al., 2021). Of interest as well are the collisions and rebounds of spheres in viscous environments (Davis et al., 1986; Gondret et al., 2002; Tan et al., 2019), the rheological properties of soft suspensions and pastes (Sekimoto and Leibler, 1993; Meeker et al., 2004), and the self-similar properties of the contact (Snoeijer et al., 2013).

In the last decade, EHD interactions have been of great interest in the material-science community with the emergence of contactless rheological methods to measure the mechanical properties of confined liquids and soft surfaces (Chan et al., 2009; Vakarelski et al., 2010; Leroy and Charlaix, 2011; Leroy et al., 2012; Villey et al., 2013; Wang et al., 2015, 2017a, 2017b; Guan et al., 2017; Wang et al., 2018; Lainé et al., 2019; Bertin et al., 2020). Typically, in such experimental systems, a spherical colloidal probe is immersed in a fluid and driven to oscillate, with a nanometric amplitude, near a surface of interest. The force exerted on the probe is measured by an atomic force microscope, a surface force apparatus or a tuning-fork microscope, and depends on the properties of both the fluid and the solid boundary.

Generally, an object that moves in a confined fluid environment experiences an enhanced drag force with respect to the bulk Stokes law, as a result of the boundary-induced flow modification (Happel and Brenner, 2012). Furthermore, near a soft wall, the hydrodynamic interactions are modified by the deformation of the boundary that they generate, yielding a nonlinear coupling. Perturbation methods, assuming a small deformation of the interface, have been employed in order to calculate the soft-lubrication interactions exerted on a free infinite cylinder immersed in a viscous fluid and near a thin compressible elastic material (Salez and Mahadevan, 2015). In particular, interesting inertial-like features have been predicted despite the low-Re-number aspect of the flow.

Perhaps the most emblematic example of soft-lubrication interaction is the non-inertial lift force predicted for a particle sliding near a soft boundary (Sekimoto and Leibler, 1993; Beaucourt et al., 2004; Skotheim and Mahadevan, 2004, 2005; Urzay et al., 2007; Urzay, 2010). It might have important implications for advected biological entities, such as red blood cells (Grandchamp et al., 2013) and vesicles (Abkarian et al., 2002). Only recently, the associated dynamical repulsion from an immersed soft interface has been studied experimentally. A preliminary qualitative observation was reported in the context of smart lubricants and elastic polyelectrolytes (Bouchet et al., 2015). Then, a study involving the sliding of an immersed macroscopic cylinder along an inclined plane pre-coated with a thin layer of gel, showed quantitatively an effective reduction of friction induced by the EHD lift force (Saintyves et al., 2016). Subsequently, the same effect was observed in the trajectories of micrometric spherical beads within a microfluidic channel coated with a biomimetic polymer layer (Davies et al., 2018), and through the sedimentation of a macroscopic sphere near a pre-tensed suspended elastic membrane (Rallabandi et al., 2018). Finally, direct measurements of the EHD lift force for two types of elastic materials have been performed at small scales, using surface force apparatus and atomic force microscopy, respectively (Vialar et al., 2019; Zhang et al., 2020).

Despite the increasing number of EHD studies involving spherical probes, the soft-lubrication interactions of a free spherical object immersed in a viscous fluid and moving near an elastic substrate still have to be calculated. In the present article, we aim at filling this gap by deriving a soft-lubrication perturbation theory, in order to compute all the forces and torque for this problem, at first order in dimensionless compliance.

The article is organized as follows. First, we introduce the soft-lubrication framework for a sphere translating near a soft planar surface, both in the normal and tangential directions. The substrate deformation is assumed to follow the constitutive response of a linear elastic semi-infinite material. Then, we perform a perturbative approach, assuming the substrate deformation to be small with respect to the fluid-gap thickness, which allows us to find the normal and tangential forces as well as the torque experienced by the sphere, at first order in dimensionless compliance. Finally, we discuss the rotation of the sphere, before providing concluding remarks. Besides, in the appendix, the EHD forces are computed analytically using the Lorentz reciprocal theorem, while the procedure introduced in the main text is reproduced for the compressible and incompressible responses of a thin material.

The system is depicted in Fig. 1. We consider a sphere of radius $a$, immersed in a Newtonian fluid of dynamic shear viscosity $\eta$ and density $\rho$. The sphere is moving with a tangential velocity $\bm{u}(t)=u(t)\,\bm{e}_{x}$ directed along the $x$-axis (by definition of the latter axis), where $\bm{e}_{j}$ denotes the unit vector along $j$. In this first part, we assume that the sphere does not rotate, i.e. the angular velocity reads $\bm{\Omega}=\bm{0}$. The sphere is placed at a time-dependent distance $d(t)$ (thus a $\dot{d}\,\bm{e}_{z}$ normal velocity of the sphere) of an isotropic and homogeneous linear elastic substrate of Lamé coefficients $\lambda$ and $\mu$, with a reference undeformed flat surface in the $xy$ plan at $z=0$. We suppose that the sphere-wall distance is small with respect to the sphere radius, such that the lubrication approximation is valid. The fluid inertia is neglected here. Specifically, we assume $\textrm{Re}\,(d/a)\ll 1$, with the Reynolds number $\textrm{Re}=\rho ua/\eta$. Furthermore, we suppose that the typical time scale of variation of the sphere velocity is much larger than the diffusion time scale of vorticity that scales as $d^{2}/(\eta/\rho)$, such that the flow is described by the steady Stokes equations. This amounts to assuming that $\lvert\dot{u}/u\rvert\ll\eta/(\rho d^{2})$ and $\lvert\ddot{d}/\dot{d}\rvert\ll\eta/(\rho d^{2})$. No-slip boundary conditions are assumed at both the sphere and wall surfaces. Finally, the system is equivalent to a sphere at rest near a wall translating with a $-(d(t)\bm{e}_{z}+\bm{u}(t))$ velocity. In such a framework, the fluid velocity field can be written as:

where $\bm{r}=(r,\theta)$ is the position in the tangential plane $xy$, $\bm{\nabla}$ is the 2D gradient operator on $xy$, $\delta(\bm{r},t)$ is the substrate deformation, and $z=h_{0}(r,t)$ is the sphere surface. Near contact, the latter can be approximated by its parabolic expansion $h_{0}(r,t)\simeq d(t)+r^{2}/(2a)$. Volume conservation further leads to the Reynolds equation:

where $h(\bm{r},t)=h_{0}(r,t)-\delta(\bm{r},t)$ is the fluid-gap thickness. In this first part, we assume that the constitutive elastic response is linear and instantaneous, and that the substrate is a semi-infinite medium, such that the deformation reads Davis et al. (1986):

We non-dimensionalize the problem through:

where $d^{*}$ and $u^{*}$ are characteristic fluid-gap distance and tangential velocity, respectively. The governing equations are then:

and:

where we introduced the dimensionless compliance:

The latter is the only dimensionless parameter in the problem. When $\kappa$ is small with respect to unity, it corresponds to the ratio between two length scales: the typical substrate deformation $\delta\sim\frac{2\eta u^{*}a(\lambda+2\mu)}{d^{*}\mu(\lambda+\mu)}$ induced by a tangential velocity $u^{*}$, and the typical fluid-gap thickness $d^{*}$. All along the article, we focus on the small-deformation regime of soft-lubrication where $\kappa\ll 1$ (Essink et al., 2021).

We perform a perturbation analysis at small $\kappa$ (Sekimoto and Leibler, 1993; Beaucourt et al., 2004; Skotheim and Mahadevan, 2004, 2005; Urzay et al., 2007; Urzay, 2010; Salez and Mahadevan, 2015; Pandey et al., 2016; Rallabandi et al., 2017; Saintyves et al., 2020; Zhang et al., 2020), as follows:

where the subscript $0$ corresponds to the solution for a rigid wall, with $H_{0}(\bm{R},T)=D(T)+R^{2}$.

We now add the rotation of the sphere, with angular velocity $\bm{\Omega}(t)$ in the $xy$ plane (see Fig. 1), to the previous translational motion. We define $\beta$ as the angle between $\bm{\Omega}$ and the $x$-axis. We stress that $\bm{\Omega}$ is not necessarily orthogonal (i.e. $\beta=\pi/2$) to the translation velocity. We discard the rotation along the $z$-axis (e.g. for a spinner), because it does not induce any soft-lubrication coupling. Finally, the system is equivalent to a purely rotating sphere with angular velocity $\bm{\Omega}(t)$, near a wall translating with a $-\bm{u}(t)$ velocity. In such a framework, the fluid velocity field at the sphere surface is $\bm{v}=-\bm{\Omega}\times a\bm{n}$, and thus $\bm{v}\simeq-\bm{\Omega}\times a\bm{e}_{z}$. All together, the fluid velocity field is modified as:

and the Reynolds equation becomes:

The problem is thus formally equivalent to the one of a sphere that is purely translating with effective velocity $\bm{\tilde{u}}(t)=\bm{u}(t)-a\bm{\Omega}(t)\times\bm{e}_{z}$. Therefore, we can directly apply the results from the previous sections, and write all the forces and torque exerted on the sphere, as:

and:

where we have invoked the force and torque induced by the rotation of a sphere near a rigid wall (Goldman et al., 1967; Urzay, 2010) and where the analytical prefactors are computed in the appendix. We stress that the expressions of the EHD forces and torque for a sphere purely translating near thin elastic substrates, as derived in appendices C and D, can be generalized to further include the sphere’s rotation by similarly following the transformation $\bm{u}(t)\rightarrow\bm{u}(t)-a\bm{\Omega}(t)\times\bm{e}_{z}$.

We developed a soft-lubrication model in order to compute the EHD interactions exerted on an immersed sphere undergoing both translational and rotational motions near various types of elastic walls. The deformation of the surface was assumed to be small, which allowed us to employ a perturbation analysis in order to obtain the leading-order EHD forces and torque. The obtained interaction matrix exhibits a qualitatively similar form as the one found for a two-dimensional cylinder moving near a thin compressible substrate (Salez and Mahadevan, 2015). In both cases, the EHD coupling is nonlinear and generates quadratic terms in the sphere velocity, thus breaking the time-reversal symmetry of the Stokes equations. In addition, original inertial-like terms proportional to the acceleration of the sphere are found – despite the assumption of steady flows. Therefore, while the quantitative details such as numerical prefactors and exponents differ in 3D and when using more realistic constitutive elastic responses, we expect that the typical zoology of trajectories identified previously (Salez and Mahadevan, 2015) will also hold for spherical objects – and will even be extended with the added degree of freedom. As such, the asymptotic predictions obtained here may open new perspectives in colloidal science and biophysics, through the understanding and control of the emerging interactions within soft confinement or assemblies.

We thank Abelhamid Maali, Zaicheng Zhang, Howard Stone and Bhargav Rallabandi for stimulating discussions.

The authors acknowledge funding from the Agence Nationale de la Recherche (ANR-21-ERCC-0010-01 EMetBrown) and from the Jean Langlois foundation.

The authors report no conflict of interest.