Elastic deformation of soft coatings due to lubrication forces

Abstract

Elastic deformation of rigid materials with soft coatings (stratified materials) due to lubrication forces can also alter the interpretation of dynamic surface forces measurements and prevent contact formation between approaching surfaces. Understanding the role of elastic deformation on the process of fluid drainage is necessary, and the case where one (or both) of the interacting materials consists of a rigid substrate with a soft coating is still limited. We combine lubrication theory and solid linear elasticity to describe the dynamic of fluid drainage past a compliant stratified boundary. The analysis presented covers the full range of coating thicknesses, from an elastic foundation to a half-space for an incomressible coating. We decouple the individual contributions of the coating thickness and material properties on the elastic deformation, hydrodynamic forces, and fluid film thickness. We obtain a simple expression for the shift in contact position during force measurements that is valid for many experimental conditions. We compare directly the effect of stratification on the out-of-contact deformation to the well-known effect of stratification on indentation. We show that corrections developed for stratification in contact mechanics are not applicable to elastohydrodynamic deformation. Finally, we provide generalized contour maps that can be employed directly to estimate the elastic deformation present in most dynamic surface force measurements. IntroductionRigid materials with soft coatings are ubiquitous in tribology, 1 microfluidic devices, 2 biomaterials, 3 or colloidal and particulate systems. 4 Under many practical settings they are employed in fluid environments where they are in close proximity to another surface. Under these conditions viscous forces due to the relative movement of the two surfaces can exert fluid pressures and cause elastic deformation (see Figure 1). More specifically, lubrication forces can lead to elastic deformation, also known as elastohydrodynamic deformation or EHD, which can prevent contact formation as fluid drains from a gap separating compliant materials. [5][6][7] If unaccounted for, EHD can lead to the misinterpretation of dynamic surface forces measurements. The surface forces apparatus (SFA) 6 or the atomic force microscope (AFM) 8 are commonly employed under dynamic conditions and can be used with soft materials. In particular, an exact description of lubrication forces is necessary to rely on dynamic surface forces measurements for the characterization of, for example, conservative surface forces, 9 fluid structure, 10 or surface slip 11 on compliant materials. In addition, recent reports show that ignoring the effect of elastic deformation can lead to a misinterpretation of the force data, contact position, and slip at the solid-liquid surfaces. 8,12 Most previous efforts to describe unsteady normal fluid drainage past an elastic boundary studied the case where the soft materials could be considered a half-space. For instance, Davis et al. studied normal e...

Influence of non-hydrodynamic forces on the elastic response of an ultra-thin soft coating under fluid-mediated dynamic loading

“…substrate behaviour approaching incompressibility), Γ, following the equation (39), is initially 1, then ∼ 1, and then 1 until γ (1 − 2ν) (RHS of equation (39)) has grown to be equal to 1 γ . As one continues to takes ν even closer to 0.5, Γ is given by equation (41) and is evidently independent of the substrate material properties, substrate deformation characteristics now exhibiting an 'incompressible substrate limit'.…”

“…In contrast, when Γ ∼ 1 γ , equations (35) to (38) are strongly coupled and not reducible to a de-coupled form. Such a coupled set of equations for the axisymmetric system considered here can be solved employing a Hankel-transformation approach [35,41,110,111], which is a scope for further generalization of current study. However, we attempt a scaling analysis for such a system, with the intent of drawing insights regarding applicability of current formulation.…”

“…This solid pressure becomes an additional unknown that has be solved for, and the incompressiblity condition, equation (40) becomes the additional equation required. Equation (40) will give the scale of Γ as presented in equation (41). We next non-dimensionalize the components of mechanical equilibrium equation and fluid-substrate interface traction-balance condition as done in appendix A, but employing the constitutive relation (43), considering Γλ(t) as the scale for u r rather than λ(t), and considering p c as the scale for the solid pressure p S .…”

“…Furthermore, there is rich literature on 'soft lubrication' too, where either the approaching object or the wall or both are deformable [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24], which finds applications in topics ranging from biotransport modelling [25][26][27][28] to tool design and analysis [29][30][31]. Such soft-lubrication setups are wide-spread in both natural and man-made world, examples being scanning probe microscope (SPM) and surface force apparatus (SFA) setups [32][33][34][35][36][37][38][39][40][41] and motion of biological entities like red blood corpuscles (RBCs) in fluidic environments [42][43][44]. With classical purely-hydrodynamic studies in hard-and soft-lubrication [45][46][47][48][49] serving as a solid foundation for further explorations, a number of later studies considered the...…”

Generalization of elastohydrodynamic interactions between a rigid sphere and a nearby soft wall

“…Seminal advancements in elastohydrodynamics (EHD) have built up crucial capabilities of predicting the force response in a well-established contactless softness characterization methodology (Steinberger et al 2008;Leroy & Charlaix 2011) and collision and rebound dynamics for approach and recession loading of a SPM probe on incompressible soft coatings (Wang, Tan & Frechette 2017;Tan, Wang & Frechette 2019). However, these studies lack sufficient generalization to accommodate large amplitudes of dynamic loading and exclusive interfacial interactions in the event that the nearby solid substrates are instantaneously located in extreme close proximity.…”