Eur. Phys. J. E (2012) 35: 5
`DOI 10.1140/epje/i2012-12005-2
`
`Regular Article
`
`THE EUROPEAN
`PHYSICAL JOURNAL E
`
`Elastic contact mechanics: Percolation of the contact area and
`fluid squeeze-out
`
`B.N.J. Persson1,a, N. Prodanov1,2, B.A. Krick3, N. Rodriguez4, N. Mulakaluri1, W.G. Sawyer3, and P. Mangiagalli5
`
`1 IFF, FZ J¨ulich, D-52425 J¨ulich, Germany
`2 Sumy State University, 2 Rimskii-Korsakov Str., 40007 Sumy, Ukraine
`3 Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville FL 32611, USA
`4 Advanced Technologies, BD Medical-Pharmaceutical Systems, 1 Becton Drive, MC 427, Franklin Lakes, NJ 07417, USA
`5 Advanced Technologies, BD-Pharmaceutical Systems, 38800 Pont de Claix, France
`
`Received 7 November 2011 and Received in final form 5 January 2012
`Published online: 26 January 2012 – c(cid:2) EDP Sciences / Societ`a Italiana di Fisica / Springer-Verlag 2012
`
`Abstract. The dynamics of fluid flow at the interface between elastic solids with rough surfaces depends
`sensitively on the area of real contact, in particular close to the percolation threshold, where an irregular
`network of narrow flow channels prevails. In this paper, numerical simulation and experimental results for
`the contact between elastic solids with isotropic and anisotropic surface roughness are compared with the
`predictions of a theory based on the Persson contact mechanics theory and the Bruggeman effective medium
`theory. The theory predictions are in good agreement with the experimental and numerical simulation
`results and the (small) deviation can be understood as a finite-size effect. The fluid squeeze-out at the
`interface between elastic solids with randomly rough surfaces is studied. We present results for such high
`contact pressures that the area of real contact percolates, giving rise to sealed-off domains with pressurized
`fluid at the interface. The theoretical predictions are compared to experimental data for a simple model
`system (a rubber block squeezed against a flat glass plate), and for prefilled syringes, where the rubber
`plunger stopper is lubricated by a high-viscosity silicon oil to ensure functionality of the delivery device.
`For the latter system we compare the breakloose (or static) friction, as a function of the time of stationary
`contact, to the theory prediction.
`
`1 Introduction
`
`The influence of surface roughness on fluid flow at the in-
`terface between solids in stationary or sliding contact is a
`topic of great importance both in nature and technology.
`Technological applications include leakage of seals, mixed
`lubrication, and removal of water from the tire-road foot-
`print. In nature, fluid removal (squeeze-out) is important
`for adhesion and grip between the tree frog or gecko adhe-
`sive toe pads and the countersurface during raining, and
`for cell adhesion.
`Almost all surfaces in nature and most surfaces of
`interest in tribology have roughness on many different
`length scales, sometimes extending from atomic distances
`(∼ 1 nm) to the macroscopic size of the system which
`could be of order ∼ 1 cm. Often the roughness is fractal-
`like so that when a small region is magnified (in general
`with different magnification in the parallel and orthogonal
`directions) it “looks the same” as the unmagnified surface.
`Most objects produced in engineering have some par-
`ticular macroscopic shape characterized by a radius of
`
`a e-mail: b.persson@fz-juelich.de
`
`curvature (which may vary over the surface of the solid)
`e.g., the radius R of a cylinder in a combustion engine.
`In this case the surface may appear perfectly smooth to
`the naked eye, but at short enough length scale, in gen-
`eral much smaller than R, the surface will exhibit strong
`irregularities (surface roughness). The surface roughness
`power spectrum C(q) of such a surface will exhibit a roll-
`off wavelength λ0 (cid:3) R (related to the roll-off wave vec-
`tor q0 = 2π/λ0) and will appear smooth (except for the
`macroscopic curvature R) on length scales much longer
`than λ0. In this case, when studying the fluid flow be-
`tween two macroscopic solids, one may homogenize the
`microscopic fluid dynamics occurring at the interface, re-
`sulting in effective fluid flow equations describing the av-
`erage fluid flow on length scales much larger than λ0, and
`which can be used to study, e.g., the lubrication of the
`cylinder in an engine. This approach of eliminating or in-
`tegrating out short length scale degrees of freedom to ob-
`tain effective equations of motion which describe the long
`distance (or slow) behavior is a very general and power-
`ful concept often used in physics, and is employed in the
`study presented below.
`
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`Eur. Phys. J. E (2012) 35: 5
`
`In the context of fluid flow at the interface between
`closely spaced solids with surface roughness, Patir and
`Cheng [1, 2] have shown how the Navier-Stokes equations
`of fluid dynamics can be reduced to effective equations
`of motion involving locally averaged fluid pressure and
`flow velocities. In the effective equation the so-called flow
`factors occur, which are functions of the locally averaged
`interfacial separation ¯u. The authors showed how the flow
`factors can be determined by solving numerically the fluid
`flow in small rectangular units with linear size of order
`of (or larger than) the roll-off wavelength λ0 introduced
`above, and by averaging over several realizations. How-
`ever, with the present speed (and memory) limitations of
`computers fully converged solutions using this approach
`can only take into account roughness over two or at most
`three decades in length scale. In addition, Patir and Cheng
`did not include the long-range elastic deformations of the
`solid walls in the analysis. Later studies have attempted
`to include elastic deformation using asperity contact me-
`chanics models as pioneered by Greenwood-Williamson
`(GW) [3], but it is now known that this theory (and other
`asperity contact models [4]) does not accurately describe
`contact mechanics because of the neglect of the long-range
`elastic coupling between the asperity contact regions [5,6].
`In particular, the relation between the average interfacial
`separation ¯u and the squeezing pressure p, which is very
`important for the fluid flow problem, is not accurately de-
`scribed by the GW model [7–9].
`The paper by Patir and Cheng was followed by many
`other studies of how to eliminate or integrate out the sur-
`face roughness in fluid flow problems (see, e.g., the work
`by Sahlin et al. [10]). Most of these theories involve solving
`numerically the fluid flow in rectangular interfacial units
`and, just as in the Patir and Cheng approach, cannot in-
`clude roughness on more than ∼ 2 decades in length scale.
`In addition, in some of the studies the measured rough-
`ness topography must be “processed” in a non-trivial way
`in order to obey periodic boundary conditions (which is
`necessary for the Fast Fourier Transform method used in
`some of these studies).
`Tripp [11] has presented an analytical derivation of the
`flow factors for the case where the separation between the
`surfaces is so large that no direct solid-solid contact oc-
`curs. He obtained the flow factors to first order in (cid:4)h2(cid:5)/¯u2,
`where (cid:4)h2(cid:5) is the ensemble average of the square of the
`roughness amplitude and ¯u is the average surface separa-
`tion. The result of Tripp has recently been generalized to
`include elastic deformations of the solids [12, 13].
`In this paper, the study of fluid squeeze-out from the
`region between two elastic solids with randomly rough sur-
`faces is presented. We focus on such high contact pres-
`sures that after long enough contact time the area of real
`contact percolates resulting in pockets of confined, pres-
`surized, fluid at the interface. The Bruggeman effective
`medium theory and the Persson contact mechanics theory
`are employed to calculate the interfacial fluid conductivity
`tensor. For anisotropic surface roughness the Bruggeman
`effective medium theory predicts that the contact area
`percolates when A/A0 = γ/(1+γ), where γ is the Peklenik
`
`number (the ratio between the decay length of the height-
`height correlation function along the two principle direc-
`tions) and A/A0 is the relative contact area (A0 is the
`nominal or apparent contact area). The main aim of the
`present work is to verify the theory predictions through
`the comparison with the results of molecular dynamics
`(MD) simulations and experiments. MD calculations have
`been carried out both for isotropic and anisotropic surface
`roughness, while the experiments consider only the sur-
`faces with isotropic statistical properties (where γ = 1).
`The paper outline is as follows. The theoretical approach
`and its application to the fluid squeeze-out are described
`in sects. 2–4 and 5, respectively. Sections 6 and 7 present
`simulations and experiments. In sect. 8 we apply the the-
`ory to the breakloose (or static) friction for prefillable sy-
`ringes. The work is closed by the concluding sect. 9.
`
`2 Anisotropic surface roughness
`
`importance have roughness
`Many surfaces of practical
`with isotropic statistical properties, e.g., sandblasted sur-
`faces or surfaces coated with particles typically bound by a
`resin to an otherwise flat surface, e.g., sandpaper surfaces.
`However, some surfaces of engineering interest have sur-
`face roughness with anisotropic statistical properties, e.g.,
`surfaces which have been polished or grinded in one direc-
`tion. The surface anisotropy is usually characterized by a
`single number, the so-called Peklenik number γ, which is
`the ratio between the decay length ξx and ξy of the height-
`height correlation function (cid:4)h(x, y)h(0, 0)(cid:5) along the x-
`and y-directions, respectively, i.e. γ = ξx/ξy. Here it has
`been assumed that the x-axis is oriented along one of the
`principal directions of the anisotropic surface roughness.
`Let us define the 2×2 matrix (we use polar coordinates
`so that the wave vector q = q(cosφ, sinφ)) [13]
`(cid:2)
`
`D(q) =
`
`dφ C(q)qq/q2
`(cid:2)
`dφ C(q)
`
`,
`
`(1)
`
`where the surface roughness power spectrum [14]
`(cid:3)
`d2x (cid:4)h(x)h(0)(cid:5)e
`1
`−iq·x,
`C(q) =
`(2)
`(2π)2
`where (cid:4)...(cid:5) stands for ensemble average, and h(x) is the
`height profile. For roughness with isotropic statistical
`properties, C(q) will only depend on q = |q| and in this
`case D(q) will be diagonal with D11 = D22 = 1/2.
`We will assume most of the time that D(q) is indepen-
`dent of q and in this case (1) is equivalent to
`(cid:2)
`
`D =
`
`d2q C(q)qq/q2
`(cid:2)
`d2q C(q)
`
`.
`
`(3)
`
`In this case, in the coordinate system where D is diagonal
`the flow conductivity matrix (defined below) σeff will be
`diagonal too. Note that TrD = D11 + D22 = 1, and in
`the coordinate system where D is diagonal we can write
`D11 = 1/(1 + γ) and D22 = γ/(1 + γ), where γ = ξx/ξy
`
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`
`Page 3 of 17
`
`is the Peklenik number. Note that D11(1/γ) = D22(γ).
`If D(q) (see (1)) depends on q we may still define (in the
`coordinate system where D(q) is diagonal) γ = −1+1/D11
`as before, but the xy-coordinate system where D(q) is
`diagonal may depend on q (in which case the rotation
`angle, ψ(q), of the x-axis relative to some fixed axis, is
`important information too, see ref. [13]). In this case γ
`will depend on q and we will refer to γ(q) as the Peklenik
`function (and ψ(q) as the Peklenik angle function). Note
`that since D(q) is a symmetric tensor and since TrD =
`1, the D-matrix has only two independent components.
`Thus, it is fully defined by the Peklenik function γ(q) and
`the Peklenik angle function ψ(q). In this paper we will
`assume that γ(q) and ψ(q) are constant.
`
`3 Fluid flow between solids with random
`surface roughness
`
`Consider two elastic solids with randomly rough surfaces.
`Even if the solids are squeezed in contact, because of the
`surface roughness there will in general be non-contact re-
`gions at the interface and, if the squeezing force is not
`too large, there will exist non-contact channels from one
`side to the other side of the nominal contact region. We
`consider now fluid flow at the interface between the solids.
`We assume that the fluid is Newtonian and that the fluid
`velocity field v(x, t) satisfies the Navier-Stokes equation
`+ v · ∇v = − 1
`∇p + ν∇2v,
`ρ
`
`∂v
`∂t
`
`where ν = η/ρ is the kinetic viscosity and ρ is the mass
`density. For simplicity we will also assume an incompress-
`ible fluid so that
`∇ · v = 0.
`We assume that the non-linear term v · ∇v can be
`neglected (this corresponds to small
`inertia and small
`Reynolds number), which is usually the case in fluid flow
`between narrowly spaced solid walls. For simplicity we as-
`sume the lower solid to be rigid with a flat surface, while
`the upper solid is elastic with a rough surface, see fig. 1.
`We introduce a coordinate system xyz with the xy-plane
`in the surface of the lower solid and the z-axis pointing to-
`wards the upper solid. Consider now squeezing the solids
`together in a fluid. Let u(x, y, t) be the separation between
`the solid walls and assume that the slope |∇u| (cid:3) 1. We
`also assume that u/L (cid:3) 1, where L is the linear size of the
`nominal contact region. In this case one expects that the
`fluid velocity varies slowly with the coordinates x and y as
`compared to the variation in the orthogonal direction z.
`Assuming also a slow time dependence, the Navier-Stokes
`equation is reduced to
`
`∂2v
`
`∂z2 = ∇p.
`(4)
`Here and in what follows v = (vx, vy), x = (x, y) and ∇ =
`(∂x, ∂y) are two-dimensional vectors. Note that vz ≈ 0 and
`
`η
`
`Fig. 1. An elastic solid (block) with a rough surface is squeezed
`(pressure p0) in a fluid against a rigid solid (substrate) with a
`flat surface.
`
`(5)
`
`(6)
`
`that p(x) is independent of z to a good approximation.
`From (4) one can obtain the fluid flow vector
`∇p.
`J = − u3(x)
`12η
`Assuming an incompressible fluid mass conservation de-
`mands that
`+ ∇ · J = 0,
`∂u(x, t)
`∂t
`where the interfacial separation u(x, t) is the volume of
`fluid per unit area. In this last equation we have allowed
`for a slow time dependence of u(x, t) as would be the case,
`e.g., during fluid squeeze-out from the interfacial region
`between two solids.
`The fluid flow at the interface between contacting
`solids with surface roughness on many length scales is
`a very complex problem, in particular at high squeezing
`pressures where a network of flow channels with rapidly
`varying width and height may prevail at the interface.
`This is illustrated in fig. 2, which shows the contact area
`(black) between two elastic solids with randomly rough
`surfaces. At high enough pressure the contact area will
`percolate, which will have a drastic influence on the in-
`terfacial fluid flow properties. Percolation corresponds to
`the moment when the narrowest channel disappears as a
`result of squeezing. It is also visible that for anisotropic
`roughness percolation occurs later in the direction of the
`roughness elongation (which is vertical in fig. 2).
`Equations (5) and (6) describe the fluid flow at the
`interface between contacting solids with rough surfaces.
`One can show that after eliminating all the surface rough-
`ness components, the fluid current (given by (5)) takes the
`form
`¯J = −σeff∇¯p,
`(7)
`where ¯p is the fluid pressure averaged over different real-
`izations of the rough surface. The flow conductivity σeff(¯u)
`is in general (for anisotropic surface roughness) a 2×2 ma-
`trix. The ensemble average of (6) gives
`+ ∇ · ¯J = 0.
`∂ ¯u(x, t)
`∂t
`Substituting (7) in (8) gives
`= ∇ ·(σ eff∇¯p) .
`∂ ¯u(x, t)
`∂t
`
`(8)
`
`(9)
`
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`Eur. Phys. J. E (2012) 35: 5
`
` 1.4
`
` 1.2
`φp
`
` 1
`
` 0.8
`
` 0.6
`
` 0.4
`
` 0.2
`
` 0
` 0
`
`γ = 2
`
`1
`
`0.5
`
` 3
` 2
` 1
`average separation / hrms
`
` 4
`
`Fig. 2. (Color online) A snapshot of the contact before per-
`colation in the x-direction (which is horizontal) for anisotropic
`roughness with Peklenik number 5/7. Red line indicates a fluid
`flow stream line. It is visible that fluid is able to flow from the
`left to the right part of the figure (or vice-versa) due to the
`presence of a narrow channel at some region of the contact.
`Inset presents the magnification of this region.
`
`Fig. 3. (Color online) The pressure flow factor φp as a function
`of the average interfacial separation ¯u, for anisotropic surfaces
`with the Peklenik numbers γ = 1/2, 1 and 2. In all cases the
`angular average power spectrum is of the type shown in fig. 4
`with H = 0.9 and the root-mean-square roughness hrms =
`10 μm.
`
`4 Fluid flow conductivity σeff
`
`where Pc(u) is a continuous (finite) function of u. Substi-
`tuting this in (10) gives
`
`1
`σeff
`
`= A
`A0
`
`1 +γ
`γσeff
`
`As was mentioned above, the fluid flow at the interface be-
`tween contacting randomly rough surfaces requires taking
`into account the presence of the network of many inter-
`connected flow channels. In a macroscopic approach this
`can be achieved through the use of the pressure flow fac-
`tor. Here we have employed the 2D Bruggeman effective
`medium theory [15–18] to calculate (approximately) the
`pressure flow factor (see also appendix B).
`For an anisotropic system, the effective medium flow
`conductivity σeff is a 2 × 2 matrix. Let us introduce a xy
`coordinate system and choose the x-axis along a principal
`axis of the D-matrix. In this case we can consider σeff
`as a scalar which within the Bruggeman effective medium
`theory satisfies the relation:
`(cid:3)
`
`1
`σeff
`
`=
`
`du P (u)
`
`1 + γ
`γσeff + σ(u) ,
`
`(10)
`
`where P (u) is the probability distribution of interfacial
`separations, and where
`
`σ(u) = u3
`12η0
`
`.
`
`(11)
`
`Fluid flow along the y-axis is given by a similar equation
`with γ replaced with 1/γ. The probability distribution
`P (u) of interfacial separations has been derived in ref. [21].
`Here we note that P (u) has a delta function at the origin
`u = 0 with the weight determined by the area of real
`contact:
`
`P (u) = A
`A0
`
`δ(u) + Pc(u),
`
`(12)
`
`1 +γ
`γσeff + σ(u) .
`This equation is easy to solve by iteration.
`In fig. 3 the pressure flow factor φp = 12η0σeff /¯u3 as
`a function of the average interfacial separation ¯u is dis-
`played for anisotropic surfaces with the Peklenik numbers
`γ = 1/2, 1 and 2 (see also below). Note that φp = 0
`for ¯u < ¯uc, where ¯uc is the average interfacial separation
`where the area of real contact percolates in the direction
`orthogonal to the fluid flow. In the Bruggeman effective
`medium theory this occurs when the area of real contact
`equals A/A0 = γ/(γ + 1). Thus for γ = 1/2, 1 and 2 the
`contact area percolates (so that no fluid flow occurs along
`the considered direction) when A/A0 = 1/3, 1/2 and 2/3,
`respectively. This explains why φp vanishes at much larger
`(average) interfacial separation (and hence smaller contact
`area) for γ = 1/2 as compared to γ = 2.
`In obtaining the results presented below we have used
`the Persson contact mechanics theory for the contact area
`A and the probability distribution P (u) (see refs. [19–21]).
`This theory depends on the elastic energy Uel stored in the
`asperity contact regions and in this paper we use the sim-
`plest version for Uel (see ref. [5]), where the γ-parameter
`(not the Peklenik number) = 1. Comparison of the theory
`predictions with numerical simulations for small systems
`have shown that γ ≈ 0.45 gives the best agreement be-
`tween theory and the (numerical) experiments. However,
`using γ = 0.45 (or γ (cid:8)= 1 in general) results in much
`longer computational time, with relatively small numeri-
`cal changes as compared to using γ = 1.
`For large (average) surface separation ¯u eqs. (5) and
`(6) can be solved exactly to leading order in (cid:4)h2(cid:5)/¯u2
`(where (cid:4)h2(cid:5) is the mean of the square of the surface rough-
`
`(cid:3)
`
`+
`
`du Pc(u)
`
`(13)
`
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`
`Page 5 of 17
`
`-18
`
`-22
`
`-26
`
`-30
`
`-34
`
`log C (m4)
`
` 3
`
` 4
`
` 6
` 5
`log q (1/m)
`
` 7
`
` 8
`
`Fig. 4. (Color online) The logarithm (with 10 as basis) of
`angular average power spectrum as a function of the logarithm
`of the wave vector. For qr < q < q1, with the roll-off wave vector
`−1 and the cut-off wave vector q1 = 108m
`−1, the
`qr = 104m
`surface is self-affine fractal with the Hurst exponent H = 0.9.
`−1 and hrms = 10 μm.
`The low wave vector cut-off q0 = 103m
`
`φp = 1 +
`
`ness amplitude h(x, y), where we have assumed (cid:4)h(cid:5) =
`0) [11, 13]
`3(cid:4)h2(cid:5)
`(1 − 3D).
`¯u2
`In appendix A we show that the Bruggeman effective
`medium theory gives the same expression for φp to leading
`order in (cid:4)h2(cid:5)/¯u2 if we identify the γ-parameter in the ef-
`fective medium theory with the Peklenik γ defined by the
`D-matrix (see sect. 2). This result shows that the parame-
`ter γ in the effective medium theory, which was introduced
`in a phenomenological way (as the ratio between the prin-
`ciple axis of an elliptic inclusion) in the effective medium
`theory (see ref. [13]), is indeed determined by the eigen-
`values of the D-matrix as discussed in sect. 2. This is a
`very important result and completes the theory for σeff
`developed in ref. [13].
`
`5 Fluid squeeze-out
`
`At high enough squeezing pressures and after long
`enough time, the interfacial separation will be smaller
`than hrms, so that the asymptotic relation (16) will no
`longer hold. In this case the relation pcont(¯u) can be cal-
`culated using the equations given in ref. [8]. Substituting
`(15) in (14) and measuring pressure in unit of p0, sepa-
`ration in unit of hrms and time in unit of the relaxation
`time
`
`4ηa2u0
`h3
`rmsp0
`
`=
`
`4ηa2
`αh2
`rmsp0
`
`,
`
`(17)
`
`τ =
`
`one obtains
`
`≈ −α
`
`−1φp(¯u)¯u3(1 − pcont),
`
`(18)
`
`d¯u
`dt
`where α = hrms/u0. In order to study the squeeze-out over
`a large time period, t0 < t < t1, it is convenient to write
`t = t0eμ (0 < μ < μ1 with μ1 = ln(t1/t0)). In this case
`(18) takes the form
`≈ −α
`
`d¯u
`dμ
`
`−1tφp(¯u)¯u3(1 − pcont).
`
`(19)
`
`This equation, together with the relation pcont(¯u), consti-
`tutes two equations for two unknowns (¯u and pcont) which
`can be easily solved by numerical integration.
`We have studied the influence of percolation on the
`fluid squeeze-out for an elastic solid with randomly rough
`surface squeezed against a rigid flat surface in a fluid with
`the viscosity η = 12 Pa s. In most of the studies the rough
`surface has the power spectrum shown in fig. 4 with the
`root-mean-square roughness hrms = 10 μm and the large
`wave vector cut-off q1 = 108 m−1. We also present some
`results for another surface with q1 = 107 m−1. The elastic
`block has rectangular shape with the width 2a = 1.84 cm
`(x-direction) and infinite length (y-direction), and the
`squeezing pressure p0 = 2 MPa. The rubber has the
`Young’s modulus E = 3 MPa and Poisson ratio ν = 0.5.
`
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`Let us squeeze a rectangular rubber block (height d,
`width (x-direction) 2a and infinite length (y-direction))
`against a substrate in a fluid. Assume that we can ne-
`glect the macroscopic deformations of the rubber block in
`response to the (macroscopically) non-uniform fluid pres-
`sure (which requires d (cid:3) a) [22, 23]. In this case ¯u(x, t)
`will only depend on time t. For this case from (9) we get
`− ¯u3φp(¯u)
`12η
`
`d¯u
`dt
`
`∂2 ¯p
`∂x2 = 0.
`It follows from this equation above that the fluid pressure
`is parabolic
`
`(cid:5)
`
`,
`
`(cid:4)
`
`1 − x2
`a2
`
`pfluid(t)
`
`3 2
`
`¯p(x, t) =
`
`where 2a is the width of the contact region (x-direction)
`and pfluid(t) the average fluid pressure in the nominal con-
`tact region. Combining the two equations above gives
`= − ¯u3φp(¯u)
`d¯u
`4ηa2 pfluid(t).
`dt
`If p0 is the applied pressure acting on the top surface of
`the block, we have
`pfluid(t) = p0 − pcont(t),
`where pcont is the (locally, or ensemble averaged) asperity
`contact pressure. If the pressure p0 is so small that for all
`times ¯u (cid:9) hrms, then in this case φp(¯u) ≈ 1. For ¯u (cid:9) hrms
`we also have [7]
`(cid:5)
`(cid:4)
`
`(14)
`
`(15)
`
`∗
`
`pcont ≈ βE
`− ¯u
`(16)
`exp
`,
`u0
`∗ = E/(1 − ν2) (here E is the Young’s modulus
`where E
`and ν the Poisson ratio), and u0 = hrms/α. The parame-
`ters α and β depend on the fractal properties of the rough
`surface [7].
`
`

`

`Page 6 of 17
`
`Eur. Phys. J. E (2012) 35: 5
`
` 0.6
`
`A/A0
`
` 0.4
`
` 0.2
`
`0.5
`
`1
`γ = 2
`
`φp = 1
`
`γ = 2
`
`1
`
`0.5
`
` 15
`
` 20
`
` 0
`-5
`
` 0
`
` 5
`
` 10
`log t (s)
`
` 15
`
` 20
`
`smooth
`
` 5
`
` 10
`log t (s)
`
`-4
`
`-6
`
`-8
`
`log uav (m)
`
`-10
`
`-5
`
` 0
`
`Fig. 6. (Color online) The relative area of real contact A/A0
`as a function of the logarithm (with 10 as basis) of the squeeze-
`out time. Calculations are for the Peklenik numbers γ = 0.5,
`1 and 2 and the same parameters as in fig. 5.
`
`φp = 1
`
`γ = 2
`
`1
`
`0.5
`
` 2
`
`pcont (MPa)
`
` 1
`
` 0
`-5
`
` 0
`
` 5
`
` 10
`log t (s)
`
` 15
`
` 20
`
`Fig. 7. (Color online) The nominal contact pressure pcon as a
`function of the logarithm (with 10 as basis) of the squeeze-out
`time calculated for the Peklenik numbers γ = 0.5, 1 and 2.
`The pink curve is the result with the fluid pressure flow factor
`φp = 1.
`
`than when γ = 1/2. In fig. 5 we also show the average in-
`terfacial separation for perfectly smooth surfaces and for
`the case where φp = 1 (which we refer to as the average-
`separation theory [23]), where the influence of the surface
`roughness is not included in the fluid flow equation. In
`this case limiting (for large time) average separation ¯u(∞)
`is determined by the roughness alone independent of the
`fluid.
`The time dependencies of the relative area of real con-
`tact A/A0 for the same systems as studied in fig. 5 are
`presented in fig. 6. For γ = 1/2, 1 and 2 the contact area
`saturates at A/A0 = 1/3, 1/2 and 2/3 but the squeeze-out
`time is the same in all cases.
`Figure 7 plots the nominal contact pressure pcon as a
`function of the logarithm of the squeeze-out time for the
`same systems as studied in fig. 5. The applied squeezing
`pressure p0 = 2 MPa is higher than the (nominal) contact
`pressure pcon and the difference p0 − pcon is carried by the
`
`Fig. 5. (Color online) The logarithm (with 10 as basis) of the
`average surface separation as a function of the logarithm of the
`squeeze-out time. Calculations are for the Peklenik numbers
`γ = 0.5, 1 and 2. The pink curve is the result with the fluid
`pressure flow factor φp = 1. For a thin rectangular rubber
`block (width 2a = 1.84 cm) with fractal-like surface roughness
`with the root-mean-square roughness amplitude hrms = 10 μm
`−1 (see fig. 4),
`and the large wave vector cut off q1 = 108 m
`squeezed against a flat rigid surface.
`
`Figure 3 displays the pressure flow factor φp as a func-
`tion of the average interfacial separation ¯u, for anisotropic
`surfaces with the Peklenik numbers γ = 1/2, 1 and 2. Note
`that the flow factor and the viscosity η enter the equation
`for the fluid squeeze-out as φp/η. Thus, φp > 1 has the
`same effect as decreasing the viscosity and hence speeds-
`up the squeeze-out. For γ = 2 we have φp > 1 except
`when the average interfacial separation ¯u becomes so small
`that the contact area is close to the percolation threshold
`where φp will vanish. Thus, at least for low squeezing pres-
`sures, where the interfacial separation never becomes so
`small that the area of real contact percolates, the fluid
`squeeze-out is enhanced by anisotropic roughness when
`the “groves” are in the direction of fluid flow. In a similar
`way, γ < 1 is equivalent to increased viscosity, and slower
`fluid squeeze-out. However, these results are only valid for
`the line-contact geometry. For a circular or square contact
`area any roughness will speed up the squeeze-out. For an
`elliptic contact area with the groves oriented along one
`of the ellipse axis, the squeeze-out may be enhanced or
`slowed down depending on the ratio between the ellipse
`axis, and the value of the Peklenik number.
`Figure 5 shows the logarithm of the average surface
`separation ¯u as a function of the logarithm of the squeeze-
`out time t for the Peklenik numbers γ = 0.5, 1 and 2. Note
`that in all cases the squeeze-out time is the same but the
`final surface separation is the largest for γ = 0.5. The rea-
`son is that for this γ the contact area percolates (in the x-
`direction) for A/A0 = 1/3, so that already when the area
`of real contact reaches this value fluid will be confined in
`the non-contact area and the rubber cannot come closer to
`the substrate as the fluid is essentially incompressible. For
`γ = 2 the contact area percolates for A/A0 = 2/3 which
`will occur at much smaller (average) surface separation
`
`Regeneron Exhibit 1110.006
`Regeneron v. Novartis
`IPR2021-00816
`
`

`

`Eur. Phys. J. E (2012) 35: 5
`
`Page 7 of 17
`
`-4
`
`-5
`
`-6
`
`-7
`
`log uav (m)
`
`-5
`
` 0
`
` 5
`
` 10
`log t (s)
`
` 15
`
` 20
`
` 25
`
`Fig. 9. (Color online) The logarithm (with 10 as basis) of the
`average surface separation as a function of the logarithm of the
`squeeze-out time. Calculations are for the Peklenik number γ =
`1. For a thin rectangular rubber block (width 2a = 1.84 cm)
`with fractal-like surface roughness with the root-mean-square
`roughness amplitude hrms = 10 μm and the large wave vector
`−1 (see fig. 4) (lower (red) curve) and q1 =
`cut off q1 = 108 m
`−1 (upper (green) curve), squeezed against a flat rigid
`107 m
`surface.
`
`(which can be observed only at high magnification) exist
`in this case. However, for “short” time the squeeze-out is
`identical for both surfaces, as is indeed expected because
`in this region of “large” surface separation the squeeze-out
`is dominated by the long wavelength roughness, which is
`the same in both cases.
`
`6 Computer simulation of percolation
`
`After the description of the theoretical approach, let us
`present the numerical simulation technique and results.
`Classical MD simulations have been carried out to verify
`the Bruggeman effective medium theory prediction that
`the contact area percolates when A/A0 = γ/(1 + γ).
`The model consists of an elastic block with a flat bot-
`tom surface which is brought into contact with a randomly
`rough rigid substrate. The latter (see fig. 10) contains
`Nx × Ny = 512 × 512 atoms which occupy the sites of a
`square lattice in the xy-plane with the lattice constant of
`a = 2.6 ˚A. Self-affine fractal topography with Hurst expo-
`nent value of H = 0.8 and the power spectrum analogous
`to the displayed in fig. 4 have been used. The isotropic ran-
`domly rough surface profile of the substrate was obtained
`using the procedure described in ref. [14], based on the
`adding plane waves with random phases. The anisotropic
`roughness was generated by stretching the isotropic one in
`one direction (corresponding to the vertical or y-direction
`in the contact pictures below) accordingly to the specified
`value of the Peklenik number. In the present study the
`values of Peklenik number equal to 1.0, 5/6, 5/7, 5/8 and
`1/2 are used.
`When two elastic solids with rough surfaces come into
`contact, the elastic deformations perpendicular to the con-
`
` 0
`log φp
`-10
`
`-20
`
`-30
`
`-40
`
`-50
`
`-60
`
`-7
`
`-6
`log uav (m)
`
`-5
`
`-4
`
`Fig. 8. (Color online) The logarithm (with 10 as basis) of the
`pressure flow factors φp as a function of the logarithm of the
`average interfacial separation ¯u for isotropic surfaces (γ = 1).
`In all the cases the angular average power spectrum is of the
`type shown in fig. 4 with H = 0.9 andh rms = 10 μm and
`−1 (red curve) and
`with the cut-off wave vector q1 = 108 m
`−1 (green curve).
`q1 = 107 m
`
`fluid. Note that for t > 1017 s the contact pressure is con-
`stant but smaller than p0 because some of the external
`load is carried by the (pressurized) fluid confined in the
`non-contact surface regions. The calculation with φp = 1
`does not account for the percolation of the contact area,
`so no pressurized confined fluid regions occur at the in-
`terface in this approximation, and after long enough time
`the full load is carried by the area of real contact. Note
`also that with φp = 1 the squeeze-out occurs faster since
`the high resistance to fluid interfacial flow which occurs
`close to the percolation threshold (because of the narrow
`flow channels) is absent in this approximation.
`Let us now study the influence of changes in the sur-
`face roughness power spectra on the squeeze-out. Figure 8
`displays the logarithm of the pressure flow factors φp as a
`function of the logarithm of the average interfacial sepa-
`ration ¯u, for a surface with isotropic roughness (Peklenik
`number γ = 1) for two values of the cut-off wave vec-
`tor q1 = 108 m−1 (red curve) and q1 = 107 m−1 (green
`curve) in the power spectrum. Note that removing one
`decade of the shortest wavelength roughness has no influ-
`ence on the pressure flow factor for large interfacial separa-
`tion because this region of fluid squeeze-out is dominated
`by the long-wavelength large-amplitude roughness compo-
`nents. However, the shortest-wavelength roughness is very
`important for small interfacial surface separations (which
`require high squeezing pressures and long enough contact
`time). Thus, removing short wavelength roughness moves
`the surface-area percolation threshold to larger interfacial
`separation.
`The dependence of the logarithm of the average surface
`