Thixotropy and Shear Thinning of Lubricated Contacts with Confined Membranes

EPJ manuscript No. (will be inserted by the editor)

Thixotropy and shear thinning of lubricated contacts with conﬁned membranes

Thomas Le Goff1, Tung B. T. To1, and Olivier Pierre-Louis1 a 1 Institut Lumière Matière, UMR5306 UniversitéLyon 1-CNRS, Universitéde Lyon 69622 Villeurbanne, France

Received: date / Revised version: date

Abstract. We have modeled the nonlinear dynamics and the rheological behavior of a system under shear containing a membrane conﬁned between two attractive walls. The presence of the membrane induces additional tangential forces on the walls that always increase the global friction. At low shear rates, the membrane exhibits chaotic dynamics with slow coarsening leading to thixotropy, i.e. to a slow decrease of the membrane-induced tangential forces on the walls. At intermediate shear rates, the membrane pro- ﬁle presents stationary periodic patterns. At higher shear rates, membrane dynamics are governed by a nonlinear evolution equation which is similar to the Kuramoto-Sivashinski equation, but with a sixth- order stabilizing term. The membrane experiences chaotic dynamics without coarsening. As a consequence of the nonlinear dynamics of the membrane at intermediate and large shear rates, the system exhibits shear-thinning.

PACS. XX.XX.XX No PACS code given

1 Intoduction

Interfaces under shear, which are at the heart of lubricated contacts, have long been recognized as a rich playground for non-equilibrium physics and pattern formation. Re- cent experimental advances have allowed one to explore the properties of nanoscale boundary lubrication, where the structuring or assembly of macromolecules controls the ultimate microscopic properties of the contact [1,2]. Progress in the understanding of these properties open Fig. 1. Schematics of a confined membrane between two at- novel avenues of research towards the tailoring of specific tractive walls. lubrication properties. In this respect, living matter has been a major source of inspiration to elaborate innovative strategies for the membranes can also be destabilized by the competing at- control of friction. Guided by selective evolution over mil- tractions of neighboring substrates, or other membranes, lions of years, nature has designed lubrication systems leading to an effective two-state adhesion potential [14, which supasses all known artificial devices, e.g. reaching 15]. Such an adhesion potential can result from the com- frictions coefficients as low as 10−3 [3,4]. Many attempts bination of attractive van der Waals forces with short have been carried out to reproduce the specific behavior of range repulsive hydration forces, leading to a minimum of biolubricating systems, using specific biological molecules the potential at some nanometers from each neighboring such as hyaluronic acid, or pointing out the crucial role of wall [16]. Other physical forces, such as osmotic forces [17] charges, e.g. on polymer brushes mimicking soft biologi- or electrostatic effects [18] may also contribute to the po- cal substrates [5–7]. Here, we wish to focus on the role of tential well near each wall. A two-state potential is also an ubiquitous component of biolubricating systems: lipidic found in biological systems when ligand-receptor adhesion membranes [8–10]. is screened at intermediate scales by glycocalyx (or by In model equations for the dynamical evolution of mem- an artificial polymer-brush mimicking glycocalyx) [19,20], branes, shear usually enters into play as a Burgers-like or when membrane adhesion is triggered by two types of nonlinear term which is known to have a stabilizing effect ligand-receptor pairs with different lengths [21]. on interface fluctuations [11–13]. However, nano-confined Here, we wish to explore the competition between the nonlinear stabilizing effect of shear and the destabilizing Send offprint requests to: influence of adhesion potentials of neighboring walls. We a Present address: Insert the address here if needed focus on the geometry shown in Fig.1. In order to simplify 2 Thomas Le Goff et al.: Thixotropy and shear thinning of lubricated contacts with confined membranes the analysis, we study a two-dimensional system where the adhesion potential. In addition, we assume that there is membrane is effectively a one-dimensional object. In addi- no slip at the membrane[22,23], so that the velocity v of tion, lipid membranes are often assumed to be incompress- the surrounding fluid is continuous at the membrane ible, so that their area is conserved in most hydrodynamic models. Here, we have not assumed that the length of the [v] = 0. (3) membrane is constant in our one-dimensional model. This The surrounding fluid is assumed to be incompressible assumption is motivated both by the sake of simplicity, ∇· v = 0, and obeys the Stokes equation: and by the fact that to a first approximation, the tran- verse dimension of the membrane, which is not considered −→ ∇p± − µ∆v± = 0, (4) here, can be considered to be a reservoir of membrane area. We solve the nonlinear dynamics of the membrane where the index ± indicates the region above or below the within the lubrication approximation, and obtain the ef- membrane. fective contribution to global friction due to the presence The walls, located at z = ±h0, move along x with the of the membrane. At small shear rates, we find that the constant velocity ±vs. We consider a no-slip boundary membrane exhibits coarsening, i.e. a slow increase of the condition at the walls, leading to characteristic wavelength as a function of time. As a consequence, the friction force decreases with time, leading to vx(x,z = ±h0,t)= ±vs, vz(x,z = ±h0,t) = 0. (5) a thixotropic behavior. At moderate shear rates, the mem- ∇· brane exhibits a frozen periodic pattern, and at large shear Integrating the incompressibility relation v =0 in rates, the membrane profile obeys Kuramoto-Sivashinsky- a rectangular domain of width dx and height 2h0 between ± ± like chaotic dynamics. These two latter regimes lead to the four points (x, h0), and (x + dx, h0), and using the shear thinning. divergence theorem together with the no-slip conditions Eqs.(5), we obtain

h0 h0 2 Model dzvx(x + dx,z)= dzvx(x,z), (6) Z−h0 Z−h0 2.1 Ingredients of the model which shows that the total flow At the continuum mechanics level of description, mem- h0 branes are different from standard interfaces between fluid j = dz vx(x,z,t) (7) Z− phases due to their bending rigidity, which is characterized h0 by the energy is constant along x, i.e. κ 2 Eb = dsC , (1) 2 Z ∂xj = 0. (8) where s is the arclength along the membrane, κ is the bending rigidity constant, and C is the curvature. When 2.2 Lubrication limit the membrane shape departs from a flat profile, the bending rigidity generates forces along the membrane normal We now consider the lubrication limit [24], where the mem- and tend to restore the planar shape. brane position h(x,t) can vary arbitrarily between the two In addition to these forces, we assume that the in- walls, but the slope ∂xh(x,t) ∼ ǫ ≪ 1 is small. The cal- teraction between the membrane and the walls produces culations are very similar to those of the case without a double-well potential U, shown schematically in Fig.1. shear[14,15]. To leading order in ǫ, the velocity compo- This potential induces additional forces which drive the nent vz is negligible, and the Stokes equations read [24] membrane towards the minima of the potential. Since we consider flat homogeneous walls, U is assumed to be a µ∂zzvx = ∂xp, 0= ∂zp. (9) function of z only, and the adhesion force on an element of membrane is oriented along the z axis perpendicular to As a consequence, denoting the quantities related to the flow above or below the membrane with and index ±, we the walls. 2 The third force acting on the membrane is the viscous have Poiseuille flows vx± = z ∂xp±/2µ + a±z + b±, where force induced by the motion of the surrounding fluid. At the six fields p±, a±, and b± depend on x but not on z. each point of the membrane, local mechanical equilibrium In order to determine these six fields, we need six rela- then reads tions. The absence of slip at the wall imposes 2 conditions [Σ] · n = f. (2) via Eq.(5). Another condition comes from the absence of slip at the membrane vx+(x,z = h) = vx−(x,z = h). Here, we have defined the fluid stress tensor Σij = µ(∂ivj+ Then, the leading order contributions of mechanical equi- ∂ v ) − pδ , where µ is the viscosity and p is the pres- j i ij librium Eq.(2) imposes two other relations ∂zvx+(x,z = sure. The brackets [.] indicate the difference between the h)= ∂zvx−(x,z = h), and fz = p+ − p−, where the force values of a given quantity just above and just below the fz acting on the membrane along z takes the form membrane, and n is the normal to the membrane. The 4 ′ force f accounts for effects of bending rigidity and of the fz = −κ∂xh −U (h). (10) Thomas Le Goff et al.: Thixotropy and shear thinning of lubricatedcontactswithconfinedmembranes 3

The sixth relation is obtain from Eq.(8), which accounts in the course of time, i.e. ∂tE ≤ 0. This quiescent regime for global conservation of mass between the two walls. has been studied recently [14,15], and leads to a frozen pe- In order to solve the differential equation Eq.(8) we need riodic metastable morphology of the membrane as shown some boundary conditions along x. For the sake of simplic- in Fig. 2, and Fig. 3(a). The wavelength of the pattern – ity and to avoid the introduction of additional parameters, physically associated to the typical size of adhesion patches, we use periodic boundary conditions in a box of size L. is that which emerges from the initial instability. It can Once the velocities vx± are known, the evolution equa- therefore be determined by a simple linear stability anal- tion for the profile h can be written from the conservation ysis, where the height of the membrane is assumed to of area below the membrane, or above the membrane exhibit small deviations δh from a flat profile: h(x,t) = h¯ + δh(x,t). Inserting this expression in Eq. (13), keep- ∂ h = −∂ j−, ∂ h = ∂ j+, (11) t x t x ing linear terms only, and going to Fourier space with where δh ∼ exp(iωt + iqx), we obtain the dispersion relation h h0 3 2 − ¯2 j− = dz vx(x,z,t), j+ = dz vx(x,z,t). (12) h0 h 2 4 ′′ ¯ ¯ Z−h0 Zh iωq = 3 q −κq −U (h) − iqγ˙ sh, (16) 24µh0 Note the two consevration laws are equivalent because j = j+ + j− is constant from Eq.(8). The resulting equation whereγ ˙ s = vs/h0 is the shear rate imposed by the motion for h reads: of the walls. The real part of iωq accounts for the rate at which the perturbations grow when ℜe(iωq) > 0, or decay 1 2 2 3 ∂ h = ∂ − h − h ∂ f when ℜe(iωq) < 0. It exhibits a maximum for q = qm, t x 24µh3 0 x z 0 corresponding to a wavelength 3j 2 2 vs − 3 ∂xh h0 − h − h∂xh, (13) 1/4 1/4 4h0 h0 3 κ λm = 2π . (17) L (−U ′′(h¯))1/4 1 2 2 j = − dxfz ∂xh (h0 − h ), (14) 2µL Z0 In order to provide physical orders of magnitudes, we con- where L is the system size. sider the specific case of an adhesion potential including Several remarks are in order. A first remarkable fea- Van der Waals attraction, following Swain et al [16]. Us- ture of the dynamics is nonlocality in space. Indeed, time- ing Hamaker constant and bending rigidity from the litera- dependent and space-independent flow rate j, which ap- ture [16–18], and assuming in addition that the membrane ¯ pears explicitly in Eq. (13), is a function of the profile is initially located halfway between the two walls (h = 0), everywhere in the system via Eq. (14). This feature is a we find λm ≈ 300nm. consequence of the incompressibility of the fluid, which imposes two conservation laws simultaneously [14]. Another important feature of Eq. (13) is its nonlinear 3.2 Friction force in the presence of shear character. Nonlinearities have three different origins. First they result from the the double-well adhesion potential U, In the following, we focus on the dynamics in the presence of shear, where we do not have a Lyapunov functional, i.e. leading to a nonlinear dependence of the force fz on h. A second source of nonlinearity are the mobilities, i.e. a functional that decreases and plays the role of the energy in Eq.(15). Indeed, shear maintains the system out- prefactors of ∂xfz, and j in Eq. (13). Due to the divergence of viscous dissipation when moving the membrane closer side equilibrium, and the wall motion continuously injects and closer to the walls, these mobilities vanish as h → mechanical work which is dissipated by the viscous liquid. Since the fluid is in the Stokes regime, the knowledge of ±h0. Shear provides a third source of nonlinearity via the the membrane profile, which determines the forces on the last term in Eq. (13). This Burger-like nonlinearity breaks membrane via Eq.(10), is sufficient to determine the ve- the x → −x symmetry, and has been reported in many locity profile inside the fluid. Hence, we also have access theoretical studies of interfaces or membranes under shear [11–to the tangential viscous forces acting at the wall in the 13]. It has a stabilizing effect on interface fluctuations, and presence of the membrane. These force are proportional a saturation effect on interface instabilities [25]. to µ∂zvx in the lubrication approximation. The average tangential force fw per unit length on one wall is

fw = µγ˙ s + fmem, (18) 3 Results L 1 3.1 Absence of shear fmem = dx h∂xhfz. (19) 2h0L Z0 In quiescent conditions when vs = 0, the last term of This force probes the global rheological behavior of the Eq. (13) vanishes and the dynamics decreases the mem- system under shear. For example, it allows one to extract brane energy the global friction coefficient νeff = fw/(2vs), or the effec- κ 2 tive viscosity of the fluid µeff = fwh0/vs in the presence E = dx (∂xxh) + U(h) (15) of the membrane. Z 2 4 Thomas Le Goff et al.: Thixotropy and shear thinning of lubricated contacts with confined membranes

Fig. 2. Snapshots of the membrane proﬁle for diﬀerent shear rates Γs.

In order to analyze and to solve numerically the membrane, we introduce dimensionless coordinates H = h/h0, 2 1/4 3/2 1/2 X = x(U0/κh0) , and T = [U0 /(24µκ )]t, where U = U0U(H), leading to

2 3 4 ′ ∂T H = ∂X (1 − H ) ∂X [∂X H + U (H)] 2 +J H − 1 ∂X H − Γ˙sH∂X H, (20) L 9 4 2 J = dX ∂X H∂X H(1 − H ). L Z0 For a given adhesion potential, the evolution equation de- pends on a single parameter, the dimensionless shear rate

1/2 1/4 ˙ µh0 κ γ˙ s Γs = 24 5/4 . (21) U0 For deﬁniteness, we choose the quartic potential

1 4 1 2 2 1 4 U4 = H − H H + H (22) 4 m 2 m 4 Fig. 4. ± Mean wavelength as a function of time for different with minima at Hm. shear rates Γ˙ . System size L = 4800. Lower panel: evolution The numerical solution of Eqs.(20) is performed with s of Fmem with time in the small shear rate regime. Hm = 0.9, both with an explicit Euler finite difference scheme, and with an implicit pseudo-spectral scheme with exponential time differencing. The explicit scheme allows The evolution of the mean wavelength λ with respect to one to ensure numerical accuracy and serves as a bench- time is plotted on Fig.4 for different shear rates. mark for the implicit scheme, and the implicit scheme allows one to integrate over longer times. For very large time the dynamics switches to a different regime. We have observed two types of asymptotic regimes shown in Fig.5. In the first one, the membrane is 3.3 Low shear rates. either frozen or the pattern translates globally at a given velocity. In the second type, a catastrophic event occurs, For low shear rates 0 < Γ˙s < 0.7 ± 0.2, the membrane dy- leading to a burst of kinks followed by a chaotic regime namics are clearly different from the case with no shear, with permanent formation and annihilation of pairs of ze- as seen in Fig. 2, and Fig. 3(b). At intermediate times the ros. After a fast increase, the average size of the adhesion membrane profile exhibits intermittent features composed patches seems to reach a steady value. The black curve of pairs of large neighboring adhesion patches on oppo- on Fig.4 corresponds to this second regime and the other site walls. Between these pairs of patches, the membrane curves to the first regime. exhibits a profile with an almost periodic pattern at a Throughout all these different regimes, the average characteristic wavelength close to that of the linear insta- size of the adhesion patches globally increases. As a con- bility. In addition, we observe coarsening events where one sequence, the force fw decreases with time as shown in adhesion patch disappears and its two neighbors merge. In Fig.4. Hence, the effective viscosity µeff (or the global this process, the number of zeros of the profile h(x,t) de- friction coefficient νeff ) decreases with time, indicating a creases, resulting in an increase of the average wavelength. thixotropic behavior. Thomas Le Goff et al.: Thixotropy and shear thinning of lubricatedcontactswithconfinedmembranes 5

Fig. 3. Dynamical regimes as a function of the shear rate Γ˙s. For the sake of clarity, we only plot the zeros of the height proﬁle as a function of time. Black symbols correspond to kinks (zeros with ∂xh > 0), and red symbols to antikinks (zeros with ∂xh < 0). (a) Γ˙s = 0, (b) Γ˙s =0.3, (c) Γ˙s =1.5 and (d) Γ˙s = 4.

∗ Γs ≈ 0.87. Interestingly, the possible heights of antikinks –i.e. a zero of h with ∂xh< 0, are actually not selected by Γ˙s, as found in a similar context in Ref. [26]. As a consequence, for positive Γ˙s, the selection of the patch height is caused by kinks, and not by antitkinks. In this regime, the periodic pattern is frozen and the force fmem is constant. As shown in Fig.6(a), the am- plitude of the steady-state force is a decreasing function of Γ˙s. The apparent scatter of the value of the force is caused by the dispersion in the number of periodic cells in our simulation box. For a box size L = 600, we find a total number of steady-state cells varying from 51 to 56. For a fixed number of cells, the force varies linearly with Γ˙s indicating that Fig. 5. Two realizations of membrane dynamics at long times ˙ 5/4 for the same value of the normalized shear rate Γs =0.3. U0 fw = µγ˙ s(1 − α) − β (23) 1/2 1/2 24h0 κ 3.4 Intermediate shear rates. with α ≈ 0.1, and β ≈ 0.03. Such a decrease of the wall force, which leads to a decrease of the friction coefficient At intermediate shear rates, the membrane reaches a peri- or the effective viscosity, is called shear thinning. odic steady-state shown in Fig. 2, and long-range order is established at the scale of the whole system, see Fig.3(c). The critical value at which the ordered periodic steady- 3.5 Large shear rates. states appear corresponds approximatively to the disap- For larger shear rates Γ˙s > Γ˙c = 3.25±0.25 the numerical pearance of steady-states with large adhesion patches. In- simulations indicate spatio-temporal chaos. In the asymp- deed, the height of the adhesion patches Hp on both sides totic limit of large shear Γ˙s ≫ 1, an expansion of the of a kink –i.e. a zero of h with ∂xh > 0, is a decreasing ˙ ˙ ˙ ∗ model equation (20) can be performed based on the ansatz function of Γs. For large enough shear Γs > Γs , the adhe- ′′ H = H¯ + H1/Γ˙s. After normalization, we obtain an equa- sion patches reach unstable heights, i.e. U (Hp) < 0. As a ˙ ˙ ∗ tion with no parameter consequence, these patches cannot exist for Γs > Γs . Us- 2 6 ing the quartic potential Eq.(22), we find numerically that ∂τ ζ = −∂χζ + ∂χζ − ζ∂χζ, (24) 6 Thomas Le Goff et al.: Thixotropy and shear thinning of lubricated contacts with confined membranes

′′ 1/4 2 3 ′′ 3/2 where χ =[−U (H¯ )] (X−Γ˙sHT¯ ), τ = (1−H¯ ) [−U (H¯ )] is quantitativelyT , more pronounced than with the poten- 2 −3 ′′ −5 4 and ζ = (1−H¯ ) [−U (H¯ )] / H. This equation is sim- tial U4. As a consequence of the strong decrease of the ilar to the famous Kuramoto-Sivashinsky equation membrane contribution to the friction force, we observe a plateau for the total friction force Fw in the intermediate 2 4 ∂τ ζ = −∂χζ − ∂χζ − ζ∂χζ, (25) regime as shown in Fig. 6(b) in black. known to exhibit spatio-temporal chaos. The numerical solution of Eq.(24) also indicates spatio-temporal chaos. 4 Discussion and conclusion Thus, the difference between the two equations, which lies in the order of the stabilizing term, does not seem to lead to a qualitative change in the dynamics, as suggested by As a final remark, the slopes of the profiles in the simula- the analogy to other nonlinear equation [27]. tions with normalized coordinates never blow up and are In this limit of large shear rates, the tangential force always of the order of one. As a consequence, the physical on the walls reads: slopes, which are of the order of ǫ ∼ h0/λm stay small at all times. This observation proves that the lubrication 2 9 ′′ 5 8 approximation is self-consistent, in the sense that if slopes h¯ (−U (h¯)) h0 A fw = µγ˙ s + 12 1 − (26) are small initially, they stay small at all times. h2 244µ3κ γ˙ 3 0 s In conclusion, confined membranes sandwiched between where U ′′(h¯) < 0, and A is a numerical constant obtained attractive walls exhibit a rich nonlinear behavior under from the simulation of Eq.(24) in a large periodic system shear, leading to complex rheological properties. Increas- of size Λ = 104 along χ: ing the shear rate, the membrane is found to exhibit end- less coarsening, frozen periodic patterns, and spatio-temporal Λ chaos. These complex nonlinear dynamics induce thixotropy 1 4 A = dχ ζ∂χζ∂χζ = 8.6 ± 0.4. (27) and shear thinning. A global diagram of the force as a Λ Z0 function of the shear rate is shown in Fig.6. Our results open new directions for the investigation of biolubrication, Note that the force on the wall is always increased by suggesting that the rich nonlinear dynamics of membranes the presence of the membrane. However, as seen from may have a non-trivial influence on the tribological prop- Eqs.(26), the membrane force f decreases for large mem erties of lubricated contacts. shear, and the system therefore also exhibits shear thinning in the large shear regime. However, since the effective We acknowledge support from the Agence Nationale −3 de la Recherche Biolub Grant (ANR-12-BS04-0008). viscosity scales asγ ˙ s , the large shear regime is different Author contribution statement: from the intermediate shear regime where a linear decrease TLG developped and was found. The prediction of Eq.(26) is shown to provide a finalized the anayltical study, performed numerical simula- good quantitative description of the asymptotic behavior tions, and wrote the preliminary version of the manuscript. of the tangential force in the inset of Fig.6(a). TBTT wrote a code for numerical numerical investiga- tions, and participated to discussions and to manuscript corrections. OPL designed the project, performed some 3.6 Changing the adhesion potential preliminary analytical study, directed TLG and TBTT, and finalized the manuscript. As seen in Fig.6, the contribution of the membrane to the global friction is small as compared to the viscous dissipation in the liquid when using the quartic potential Eq.(22) References with Hm = 0.9. In order to investigate the possibility of a larger contribution of the membrane, we chose a simple 1. A. Erdemir, Tribology International 38, 249 (2005), destabilizing quadratic potential boundary Lubrication 2. B. 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Fig. 6. Tangential forces on the walls. (a) Normalized membrane contribution Fmem as a function of Γ˙s for U4 and Hm =0.9. The arrow towards the bottom indicate that friction force decreases with time for low shear rates in the coarsening regime. The ˙ 3 inset in log-log scale with the analytical prediction Fmem ∝ 1/Γs at large shear rates. The gray zones indicate the transition region between diﬀerent regimes. (b) Total friction force Fw as a function of Γ˙s. Red points are obtained with U4 and Hm =0.9, 2 1/2 and black points with U2 and Hm =2.4 . The green line represents the friction force without membrane.

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