Journal of Colloid and Interface Science 221, 1–12 (2000) Article ID jcis.1999.6558, available online at http://www.idealibrary.com on
Elastohydrodynamic Collision of Two Spheres Allowing Slip on Their Surfaces Olga I. Vinogradova∗, †,1 and Fran¸cois Feuillebois‡ ∗ Institut f¨ur Physikalische Chemie, Universit¨at Mainz, Jakob-Welder-Weg 11, 55099 Mainz, Germany; †Laboratory of Physical Chemistry of Modified Surfaces, Institute of Physical Chemistry, Russian Academy of Sciences, 31 Leninsky Prospect, 117915 Moscow, Russia; and ‡Laboratoire de Physique et M´ecanique des Millieux H´et´erog`enes, ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France Received February 9, 1999; accepted September 30, 1999
numerical techniques. It was assumed that the no-slip boundary condition is valid and that any surface interactions can be ignored. The joint effect of colloidal and hydrodynamic forces on deformation and coagulation was considered in studies (6, 7) again assuming that the no-slip boundary condition is fulfilled and restricting the colloidal interactions to those described by the DLVO theory. These authors found that the colloidal forces primarily come in to play after the particles have stopped, so that they can be ignored during the approach stage. Although these theoretical works are important contributions to the subject, they all ignore the effect of slippage on the film drainage between elastic interfaces. On the other hand, there is a considerable body of research based on lubrication asymptotics and pertaining to thinning rates between (rigid) interfaces allowing slip (8–10). Thus, many systems such as a polymer melt (11, 12) flowing over a solid or water flowing past a hydrophobic solid (13, 14) remain beyond the scope of their application.2 Reference (15) is especially relevant here. These authors extended the analysis by considering the effect of slippage on the head-on interaction of two elastic spheres. The model was developed for the simplest case of spheres with equal slip length on both surfaces and small deformation. The main focus was on the determination of the criteria which govern whether two spheres will stick or rebound. Based on the fact that slippage makes the collision of rigid particles possible even in the absence of attractive colloidal forces, the authors coupled the lubrication flow in the gap between slippery surfaces with the model predicting the minimum incoming kinetic energy that would allow colliding particles to escape (rebound).
Our goal is to study theoretically the effect of deformation on the collision of two solid spheres allowing slip on their surfaces. The deformed shape of the solid surface is determined via an asymptotic technique assuming that deformation is small compared with the separation between the surfaces. It has previously been shown that the slippage makes collision possible even without any surface attractive force. Here we demonstrate that even a small amount of deformation can preclude spheres from coagulation. °C 2000 Academic Press Key Words: deformation; lubrication; slippage; collision.
I. INTRODUCTION
Sedimentation, filtration, coagulation, and other processes often occur under conditions of relative motion of solid particles in a liquid. In many of these applications, a key role is played by collisions between two interacting particles. If particles are elastic, then at small interparticle separations surface and hydrodynamic forces can cause their deformation. In recent years several theoretical papers have been concerned with deformation of elastic spheres. We mention below what we believe are the most relevant contributions. A few reports were devoted to the problem of elastic deformation of solids under the action of surface forces. Early solutions of the elastic equations include the asymptotic results for repulsive interactions (1, 2) and numerical results for attractive Lennard–Jones interactions (3). A recent study (4) suggested a numerical solution for three types of surface forces, namely molecular attractive, electrostatic repulsive, and oscillatory (or solvation) forces. In all these works it was assumed that the surface approach is so slow that the hydrodynamic effects can be ignored. However, in many situations it proves to be incorrect to neglect the approach dynamics. The dynamic deformation of elastic spheres during their inertial collision was studied in reference (5). The deformed shape of the solid surfaces and the pressure profile in the fluid layer separating these surfaces were determined via asymptotic and
2 It is perhaps worthwhile to comment here on a connection to a problem that arises in gas film lubrication. In the latter case “the Maxwell slip flow approximation,” which gives a correction to continuum flow, is valid only for values of the Knudsen number that are significantly different from zero, but still small (0 < Kn ¿ 1). We remark that Kn can be treated as a ratio of the slip length to the gap if the accomodation coefficient is close to unity (diffuse reflection). In other words, gas film lubrication would allow the application of the slip model only for small amount of slippage and would require the use of the kinetic theory of gases or molecular gas dynamics for a large slip. Such a restriction should not be applied for the case of slippage of polymeric fluid and/or hydrophobic slippage which seem to be not molecular in their origin (for a recent review see Ref. (14) and references therein).
1 To whom correspondence should be addressed. E-mail: olgav@wintermute. chemie.uni-mainz.de,
[email protected].
1
0021-9797/00 $35.00
C 2000 by Academic Press Copyright ° All rights of reproduction in any form reserved.
2
VINOGRADOVA AND FEUILLEBOIS
Our present study seeks to address issues similar to those discussed in Ref. (15) but we go further by considering arbitrary slip lengths on the surfaces of the approaching bodies. Beside that we show that deformation and slippage, acting in opposite directions, have a combined effect on coagulation. The possibility for a collision to occur in a finite time due to tangential mobility of a liquid on the particle surfaces is inhibited, because this also leads to the increase in deformation, and as a result precludes the elastic particles from coming into contact. Our paper is arranged as follows: In Section II we present the governing equations for a flow of liquid confined between elastic solids allowing slip on their surfaces. The asymptotic solution for small deformation is given in Section III. The results of our calculations are presented in Section IV. We conclude in Section V with a discussion of some aspects of the interaction of elastic solids. II. GOVERNING EQUATIONS
In this section we define our system and summarize earlier relationships which are pertinent to the present analysis. The generalization of a lubrication approach to elastic slippery surfaces is also described. Consider two elastic spherical particles with radii R1 and R2 , immersed in a Newtonian liquid. Our aim is to determine the shape of the deformed surfaces as a function of time and the distance from the line connecting the centers of the spheres. We also have to find the relative approach velocity of surfaces as a function of time. We consider a situation in which the gap between the spheres is small compared to the smaller of their radii. The rugosity of surfaces is ignored. The particles move along the axis connecting their centers with velocities v1 and v2 . For the initial condition at t = 0, we specify that the spheres start with a gap h 0 between their undeformed surfaces and with a relative velocity v0 . We require that the spheres are rigid enough so that the surfaces deform only within a small area near the axis of symmetry. This latter restriction is reasonable for most solid particles. The deformed and undeformed surfaces of the two spheres are sketched in Fig. 1. The undeformed spherical
surface H˜ can be approximated by a paraboloid r2 , H˜ (r, t) = h(t) + 2R
[2.1]
where h(t) is a distance between the undeformed spheres and R is the reduced radius R1 R2 /(R1 + R2 ). The deformed gap profile can be locally given as H (r, t) = h(t) +
r2 + w(r, t), 2R
[2.2]
where w(r, t) = w1 (r, t) + w2 (r, t) is the sum of deformations of the two surfaces from their original shape. To determine the deformation we shall follow the ideas of the Hertz contact theory of linear elasticity (16) and also the developments of authors in (1, 3, 5) who studied the deformation due to the action of surface and hydrodynamic forces. For completeness we mention briefly some of the relationships. The essence of the theory is that an applied normal force distributed over the surface will cause it to deform. 3 Provided that this deformation is small, it can be determined by integrating the surface stress distribution multiplied by Green’s function over the area subjected to the stress, where the appropriate Green’s function is the fundamental solution of the linear-elasticity equation for an applied point force of unit magnitude. For our problem, the formulation of this integral is (5) Z 4θ ∞ P(y, t)φ(r, y) dy, [2.3] w(r, t) = π 0 where P(r, t) is a pressure inside the gap. In general, it is equal to the sum of the hydrodynamic p(r, t) and the disjoining 5(r, t) pressures. However, it has previously been shown that the surface interactions primarily come into play after the particles have stopped (6, 9) and much less than the hydrodynamic pressure when particles are in relative motion. Hence, it is possible to solve an inertial collision problem ignoring (in the first-order approximation) all the surface forces, i.e., assuming P(r, t) = p(r, t). The parameter θ in [2.3] is defined as θ=
1 − ν22 1 − ν12 + , E1 E2
[2.4]
where ν1,2 is Poisson’s ratio for sphere 1, 2, and E 1,2 is Young’s modulus of elasticity for sphere 1, 2. Green’s function kernel is given by · ¸ 4r y y K , [2.5] φ(r, y) = y +r (r + y)2 where K is the complete elliptic integral of the first kind.
FIG. 1. Schematic of the deformation of two colliding elastic spheres in a viscous liquid. The solid curves denote the actual deformed surfaces and the dashed curves denote the undeformed surfaces.
3
It is straightforward to show that in the case of a lubrication theory the tangential forces can safely be neglected and deformation is due to a normal force.
3
ELASTOHYDRODYNAMIC COLLISION
The deformed shape of the sphere surfaces cannot be determined without knowledge of the pressure profile in the liquid layer between the surfaces. Although the lubrication approach has been employed in many studies, it is appropriate to clarify some details about the solution of the equations of motion, because they are applied now for a more complex case of slippery elastic solids. The flow of liquid in the gap between the bodies should satisfy the Navier–Stokes equations, which can be substantially simplified, since the lateral component of the velocity field is large compared with the normal component. The classical lubrication approximation then gives µ
∂p ∂ 2 vr ∼ , ∂z 2 ∂r
[2.7]
where vz is the projection of the liquid velocity on the axis z forwarded along the line connecting the centers of the spheres. The boundary conditions express the slippage of liquid over the hydrophobic surfaces. Assume that for one surface the value of slip length is equal to b, while for the other surface it is b(k + 1), where k can have values between −1 and ∞ (10).4 Therefore, the boundary conditions take the form ∂vr At z = 0, vz = 0, vr = b ∂z ¶ µ ∂w ∂H r + = , At z = H, vz − vr R ∂r ∂t vr = −b(k + 1)
∂vr . ∂z
α(k, H ) =
By integrating the continuity Eq. [2.7] with [2.8]–[2.9], and using [2.10], we derive that the hydrodynamic pressure in the narrow gap between the deformed spheres with the slip lengths b and b(k + 1) can be found from the equation ∂H 1 ∂ = ∂t 2µr ∂r
·µ ¶µ 3 ¶¸ ∂p H H2 r − α(k, H ) − bH α(k, H ) , ∂r 3 2 [2.11]
[2.8]
[2.9]
where p(r, t) is the hydrodynamic pressure profile in the liquid layer which, to leading order, does not vary at any given time across the width of the narrow gap. Equation [2.11] is exact and valid for any deformation, however large. We note that its form coincides with the equation for rigid spheres (10). However, in our case the thickness H (r, t) is defined by Eq. [2.2] and includes deformation. The consequence from this fact is that, in the general case, and in contrast to rigid surfaces, the pressure cannot be expressed as a function of the gap profile H . III. ASYMPTOTIC SOLUTION
A. Pressure Profile The solution of the system of Eqs. [2.2], [2.3], and [2.11] in general requires a numerical method. However, for small deformation (w ¿ h), these equations can be solved by an asymptotic method. In this case, in the first approximation the deformation can be determined via the pressure profile in the absence of de˜ t), and the velocity of the relative motion of the formation p(r, surfaces −∂ H/∂t is approximately equal to the relative velocity of the centers of mass of the two spheres v (5). By setting w = 0 in [2.2] and then solving [2.11] for the pressure profile, we find that ˜ t) = p(r,
The solution of Eq. [2.6] taking into account the boundary conditions [2.8] and [2.9] leads to 1 ∂p 2 · [z − zα(k, H ) − bα(k, H )], vr = 2µ ∂r
H (H + 2b(1 + k)) . H + b(2 + k)
[2.6]
where µ is the dynamic viscosity and vr is the radial liquid velocity. We remark that although the Reynolds number Re is not small, the inertia of liquid was neglected in our analysis (although we will retain the inertia of particles later). This is justified provided that Re · h 0 /R ¿ 1. The continuity equation is 1 ∂ ∂vz + · (r vr ) = 0, ∂z r ∂r
where
3µRv · p∗ , H˜ 2
where the dimensionless function p ∗ is given by [2.10]
4 The lower value of k corresponds to the situation of the interaction of a slippery particle with a particle that does not allow slip. The upper value of k is characteristic of the situation where a slippery particle interacts with a frictionless sphere. The condition of zero friction is usually valid for a bubble surface, which is certainly not elastic, and it is difficult to say now how realistic this situation is for the solid/liquid interface. However, there are some first reports about a liquid flow between two solids that exhibit full-slip (zero friction) (17, 18).
(i) k → −1, ¶¸¶ µ · µ 1 4b H˜ H˜ 1+3·2· 1− ln 1 + p = 4 4b 4b H˜ ∗
(ii) k → ∞, · µ ¶¸ H˜ H˜ 1 3b 1− ln 1 + p = ·2· 4 3b 3b H˜ ∗
[3.1]
4
VINOGRADOVA AND FEUILLEBOIS
(iii) k 6= −1, k 6= ∞ ¶ · · µ H˜ H˜ x1 x2 − x1 bx2 p = 2· + ln 1 + b x2 x3 b(x3 − x2 ) x22 H˜ µ ¶¸¸ x3 − x1 bx3 − ln 1 + x32 H˜ ∗
with x1 = 2 + k x2 = 2(2 + k + x3 = 2(2 + k −
p p
This is the maximum value of the integral, which corresponds to the maximum deformation. For k → 0 the last equation can be substantially simplified √ 2 p 2π I (β, 0) = (1 + 3β − 1 + 6β), 2 36β
1 + k + k2) 1 + k + k 2 ).
These results are certainly equivalent to the solution for rigid spheres (10). An especially important case to mention is that of k → 0 corresponding to the situation b1 = b2 = b, which has previously been analyzed for both rigid (8, 9) and elastic (15) particles. In that case, the expression for p ∗ is transformed to (8–10, 15) · µ ¶¸ 6b H˜ H˜ 1− ln 1 + p =2· 6b 6b H˜
π I (β, ∞) ∼ 8ξβ
B. Deformation Profile Putting [3.1] into [2.3], we obtain a deformation profile due to nonuniform hydrodynamic pressure along the liquid film 12µθv π
µ ¶3/2 R I (β, ξ ) , h
I (β, ξ ) = 0
∞
φ(ξ, η) · p ∗ (η, β) dη. (1 + η2 /2)2
[3.3]
Here we used the scaling y b r ξ = √ , η= √ , β= . h Rh Rh In general, this integral must be performed numerically. However, analytic expressions may be obtained in the limits of small and large values of the radial position. Taking into account that φ(ξ, η) = π/2 at ξ = 0, we have (i) k → −1 √ 2 p 2π 2 (3 + 6β + 2β − 3 1 + 4β) I (β, 0) = 4 · 16β 2 (ii) k → ∞ √
I (β, 0) =
p 2π 2 (2 + 3β − 2 1 + 3β) 2 4 · 18β
µ
· ¸ ¶ 3 3 1 − +β + 1+ ln(1 + 4β) 2 2 4β
(ii) k → ∞ π I (β, ∞) ∼ 12ξβ
µ
· ¸ ¶ 1 −1 + 1 + ln(1 + 3β) 3β
[3.2]
with Z
and when β → 0, this limiting expression for I (β, 0) transforms √ to the known result for hydrophilic surfaces I (0, 0) → π 2 2/8 (5, 19). For large distances r from the axis, we have η/ξ ¿ 1 and φ(ξ, η) ∼ π/2 · η/ξ, so that the integral I (β, ξ ) behaves asymptotically as (i) k → −1
∗
w(r, t) =
(iii) k 6= −1, k 6= ∞ √ 2µ √ √ (x1 − x2 ) βx2 + 1 (x1 − x3 ) βx3 + 1 2π I (β, 0) = + β2 x22 (x2 − x3 ) x32 (x3 − x2 ) ¶ βx1 x1 x2 + x1 x3 − x2 x3 + . + 2x2 x3 x22 x32
(iii) k 6= −1, k 6= ∞ µ π (x1 − x2 )(βx2 + 1) ln(1 + βx2 ) I (β, ∞) ∼ − ξβ βx22 (x2 − x3 ) −
x1 (x3 − x1 )(βx3 + 1) ln(1 + βx3 ) − 2 x2 x3 βx3 (x2 − x3 )
¶
which for k → 0 reduces to µ · ¸ ¶ 1 π −1 + 1 + ln(1 + 6β) . I (β, ∞) ∼ 6ξβ 6β When β → 0 the last expression gives I (0, ∞) ∼ π/2ξ (5). C. Velocity of Particles To complete the formulation of the problem, we also require the kinematic equations which describe the relative motion of the undeformed surfaces of the solid spheres dh = −v(t), dt dv = −F(t), m dt
[3.4] [3.5]
5
ELASTOHYDRODYNAMIC COLLISION
where v(t) = v1 (t) − v2 (t) is the relative velocity of the centers of mass of the two spheres, m is the reduced mass m 1 m 2 / (m 1 + m 2 ), and F(t) is the total force acting between the two spheres. For the case we are interested in, this is just a hydrodynamic force Fh . From Eq. [3.1], the hydrodynamic force on the spheres is 6π R 2 µv · f ∗, F˜ h = h
(iii) k 6= −1, k 6= ∞ x1 v = x2 x3 ∗
(i) k → −1 µ ·µ ¶ ¸¶ 1 1 1 1+3·2· 1+ ln(1 + 4β) − 1 f∗ = 4 4β 4β
f∗ =
1 1 ·2· 4 3β
1+
1 3β
¶
Here, we introduced β0 = b/ h 0 = β · δ with δ = h/ h 0 . The Stokes number is defined as
¸ ln(1 + 3β) − 1
St =
(iii) k 6= −1, k 6= ∞ f∗ = −
¸
· ¸ (x2 − x1 ) 1 + 2βx2 β − ln(1 + βx2 ) 1 + β0 (x2 − x3 ) β 2 x22 · ¸ (x3 − x1 ) 1 + 2βx3 + ln(1 + βx3 ) 1 + (x2 − x3 ) β 2 x32 · ¸ 1 + 2β0 x2 (x2 − x1 ) ln(1 + β0 x2 ) 1 + + (x2 − x3 ) β02 x22 · ¸ 1 + 2β0 x3 (x3 − x1 ) ln(1 + β0 x3 ) 1 + . − (x2 − x3 ) β02 x32
with
·µ
1 1 − β β0
+ ln
[3.6]
(ii) k → ∞
·
· 2x1 2 (βx2 + 1)(x2 − x1 ) − 2 βx2 x3 β (x3 − x2 ) x22
¸ (βx3 + 1)(x3 − x1 ) × ln(1 + βx2 ) − ln(1 + βx3 ) . x32
mv0 6π µR 2
and provides a measures of the inertia of particles relative to the viscous forces. It is evident that the inertial collision we consider here is possible only when St is not small compared to unity. For k → 0 ¶ µ µ ¶ β(6β0 + 1) 1 1 v = ln − 1− 2+ ln(1 + 6β0 ) β0 (6β + 1) 6β0 6β0 µ µ ¶ ¶ 1 1 1− 2+ ln(1 + 6β) , [3.8] + 6β 6β ∗
From [3.1], [3.2], and [3.3] we see that the pressure and deformation profiles depend parametrically on the values of v and h, which are functions of time. Eliminating time from Eqs. [3.5] and [3.4], using [3.6] and integrating, we find v v∗ =1− , v0 St
which gives [3.7] v ∗ = − ln δ
where (i) k → −1 β 3 4β0 + 1 + ln v ∗ = ln β0 4 4β + 1 · µ ¶ ¸ 1 3 1− 2+ ln(1 + 4β) + 4 · 4β 4β ¸ · µ ¶ 1 3 1− 2+ ln(1 + 4β0 ) − 4 · 4β0 4β0 (ii) k → ∞ ¶ µ µ µ ¶ 1 β(3β0 + 1) 1 1 v∗ = ln − 1− 2+ ln(1 + 3β0 ) 4 β0 (3β + 1) 3β0 3β0 µ µ ¶ ¶¶ 1 1 1− 2+ ln(1 + 3β) + 3β 3β
when β0 , β → 0. This is a known solution for the no-slip limit: the velocity decreases linearly with ln δ and vanishes at δ = exp(−St). At this point, the dissipative forces of the viscous fluid stop the inertial-driven motion of the spheres. For a β0 value below a certain critical value (which is a function of St and k) or for a St value below a certain critical value (which is a function of β0 and k), the particles stop at a gap distance given by the solution of Eq. [3.7] for δ with v set equal to zero (or v ∗ = St). By setting both h and v equal to zero in Eq. [3.7] we can determine the critical Stokes number, St ∗ (β0 , k), corresponding to the critical β0 value which just allows the particles to touch: (i) k → −1 ¶ µ ln δ →∞ St = − 4 δ→0 ∗
6
VINOGRADOVA AND FEUILLEBOIS
FIG. 2. Integrals I (β, ξ ) calculated in the no-slip limit (solid curve) and for particles allowing slip (dashed curves from top to bottom β = 10−2 , 10−1 , 1, 10), I (β, 0) (×), and I (β, ∞) (+): k = −1 (A), 0 (B), 20 (C), ∞ (D).
(ii) k → ∞ St ∗ =
¸¸ · µ ¶ · µ ¶ 1 1 1 1 ln 1 + − 1− 2+ ln(1 + 3β0 ) 4 3β0 3β0 3β0
(iii) k 6= −1, k 6= ∞ St ∗ =
µ µ ¶ ¶ x2 − x1 1 1 x3 − x1 ln 1 + ln 1 + − x2 − x3 β0 x2 x2 − x3 β0 x3 µ · ¶ 1 (x2 − x1 ) ln(1 + β0 x2 ) x1 1 2+ − − β0 x2 x3 x2 (x2 − x3 ) β0 x2 µ ¶¸ 1 (x3 − x1 ) ln(1 + β0 x3 ) 2+ . + x3 (x2 − x3 ) β0 x3
An important point to note is that our expression for the force in the case of particles with the same slip length is different from
that used in Ref. (15). The authors of Ref. (15) have used an approximate expression for arbitrary separation,5 but in the current paper we use the exact expression for small gap. In principle, this should lead to an only slightly different expression for the particle velocity, because a “uniformly valid” form of [15] for the resistance is only very slightly different from the lubrication form for small gaps. Indeed, the addition of the Stokes drag force should result only in the additional term h 0 /R · (1 − δ) in the expression for v ∗ , and, correspondingly, h 0 /R in the expression for St ∗ . However, the factor h 0 /R, which is the small paramerer of the problem, was lost in Ref. (15). That’s why some of their results are different from ours (see below). This is, however, only a quantitative difference.
5
At the same time the lubrication expression for pressure is applied.
7
ELASTOHYDRODYNAMIC COLLISION
IV. RESULTS AND DISCUSSION
The role of slip in determining the deformation becomes clear from the analysis of Eq. [3.2]. On the one hand, as has been shown in Ref. (10), slippage decreases p ∗ . Hence it leads to a decrease in I (β, ξ ). On the other hand, slippage increases the particle approach velocity. These factors, acting in opposite directions, have a combined effect on deformation. The integral I (β, ξ ) and the asymptotic results for different values of k are plotted in Fig. 2. This integral has been calculated numerically.6 The maximum value of I (β, ξ ) occurs at r = 0, and it is largest in the no-slip limit but decreases with increasing β and k. Analytic expressions obtained in the limits of small (I (β, 0)) and large (I (β, ∞)) values of the radial position can be presented as the product of the formulas for the no-slip limit (I (0, 0) and I (0, ∞)) and some dimensionless corrections for slippage. Figure 3 displays these corrections as a function of h/b obtained for different values of k. We note that due to slippage the behavior of the integral I (β, ξ ) is very rich compared with the no-slip case. Clearly, the reason for this is that the slip lengths of the bodies define two additional length scales of the problem. This leads to a complex hydrodynamic behavior of the thin liquid layer when its thickness becomes comparable with the slip lengths. A typical plot of the particle velocity as a function of separation for various values of β and k is shown in Fig. 4, and the critical Stokes numbers as a function of β0 and k are plotted in Fig. 5. For the coordinate pair below the curves, the spheres do not make contact: their kinetic energy is completely dissipated by viscous forces. We note that when there is at least one surface with a no-slip boundary condition (either β0 = 0 or k → −1), then St ∗ → ∞ is required for the (rigid) particles to make contact. We also remark that at small β0 the value of St ∗ decreases linearly with ln β0 : if k → ∞ µ ¶ 1 3 ∗ St = − ln 3 − ln β0 4 2 if k 6= ∞ St ∗ =
3 (x1 − x2 ) ln x2 − (x1 − x3 ) ln x3 + − ln β0 . 2 x2 − x3
For a given St and β0 above the curves, the similar rigid spheres have sufficient inertia to make a contact. Below we demonstrate that the elastic spheres exhibit different behavior. 6
We note that φ(r, y) is logarithmically singular at y/r → 1 φ(r, y) ∼
1 8 ln . 2 |1 − y/r |
The difficulty in performing the numerical integration was overcome in the usual way by subtracting the singularity. Also, for large η the integration was performed analytically using the appropriate asymptotic expansions for the pressure profile and for the kernel φ(r, y).
FIG. 3. The correction for slippage in the deformation integral I (0, ξ ) at ξ → 0 (A) and ξ → ∞ (B) vs h/b for different k (semilogarithmic scale). The solid curves from left to right are the calculation results for k → −0.99, 101 , 102 , 103 , 104 , 105 , and 106 . The dashed, dotted, and dash–dotted curves show the calculation results for k → −1, 0, and ∞, respectively.
The conditions under which the ratio of the deformation to the gap remains small may be found by substituting Eq. [3.7] for the relative velocity and the equation for I (β, 0) into Eq. [3.2], and then dividing the result by h: w(0) = ² δ −5/2 h
µ
v v0
¶ I (β, 0).
[4.1]
Here, similarly to (5), we introduced the dimensionless elasticity parameter defined by ²=
12θ µv0 R 3/2 5/2
π h0
.
This parameter provides a measure of the tendency of the solids to deform. Its value must be small compared to unity in order for
8
VINOGRADOVA AND FEUILLEBOIS
FIG. 4. The particle velocity as a function of separation (between rigid spheres) for St = 5 in the no-slip limit (solid curve) and for particles allowing slip (dashed curves from bottom to top β0 = 10−3 , 10−2 , 10−1 , 1, 10): k = −1 (A), 0 (B), 20 (C), ∞ (D).
the deformation to be small. One can expect that for the majority of the systems the typical values for ² are in the range 10−7 –10−5 . Below we analyze Eq. [4.1] in order to understand the qualitative behavior of the particles and show that the deformation of the particles is entirely determined by the behavior of the particle approach velocity when the gap is small. When St and β0 are below the critical values (see Fig. 5), the integral I (β, 0) is finite, and particles exhibit the following behavior. The spheres start to approach each other. As the gap between surfaces decreases, the pressure in the liquid layer increases. The pressure causes the deformation (flattening) of the particles. However, the pressure also causes the spheres to slow down, which eventually leads to a decrease in the pressure. As a result, at some moment of time the deformation of the surfaces reaches a maximum7 and then a relaxation occurs. For example,
7 The reduced gap h/ h corresponding to the maximum deformation does 0 not depend on ² and is only a function of St, k, and β0 .
in the no-slip limit the maximum value of w/ h occurs at (5) µ δ = exp
2 − St 5
¶
and is equal to √ µ ¶ ²π 2 2 5St w(0) = · exp −1 . h 20St 2
[4.2]
In Fig. 6 the computed values of w(0)/ h are plotted as a function of δ. We see that with an increase in β0 and k the value of deformation maximum increases. For example, in the no-slip limit in order for the deformation to be small we should require µ ¶ 5St 20St . ² ¿ √ exp 1 − 2 π2 2 For the slippery particles the condition for ² is stronger. The
9
ELASTOHYDRODYNAMIC COLLISION
FIG. 5. Critical Stokes number which allows spheres to make contact as a function of β0 . From top to bottom k → −0.99, 0, 101 , 102 , ∞. The solid curves are the exact results; the dashed lines show the asymptotic results for small β0 .
gap corresponding to this maximum becomes smaller with an increase in β0 and k. However, the thin liquid layer always prevents the surfaces from actually touching. We also remark that at large surface separation the deformation of the slippery particles is less than that for the no-slip limit. In the case of critical St and β0 the impact velocity for rigid spheres turns to zero as ¶ µ 1 δ 3β0 v ∼ · ln v0 2St ∗ 3β0 δ for k → ∞, and for k 6= ∞ the asymptotics is ¶ · µ 2 x1 − x2 δ x2 β0 v ∼ ∗· ln v0 St x2 − x3 β0 x2 δ µ ¶¸ x3 β0 x3 − x1 δ ln . + x2 − x3 β0 x3 δ In this situation, the integral I (β, 0) turns to zero as I (β, 0) ∼
π 2δ √ 12 2β0
for k → ∞ and as I (β, 0) ∼
x1 π 2 δ √ x2 x3 2β0
for k 6= ∞. It follows then that w(0) → 0 at h → 0, but the func-
tion w(0)/ h diverges as w(0) ∼ −δ −1/2 ln δ. h
[4.3]
The case of St and β0 above the critical values is characterized by a finite nonzero value of v/v0 at h → 0. As a result at h → 0 the deformation w(0) → ∞, and the relative deformation diverges as w(0) ∼ δ −3/2 . h
[4.4]
This means that deformation precludes spheres from coming into contact (see Fig. 6), a conclusion which is similar to that made first for the deformable drops and bubbles (20, 21). Thus, we stress that for both critical and supercritical St and β0 the physical contact and the consequent rebound of the elastic particles suggested in Ref. (15) are impossible in the framework of their (and our) original assumption of small deformation. The calculations of the elastohydrodynamic deformation in (15) for identical (k = 0) slippery particles give that at St = St ∗ the curves w(0)/ h vs δ exhibited nonmonotonic behavior. In our results, the ratio of deformation to the gap for critical St was always monotonic. We use absolutely identical expression for pressure, so that the expression for deformation is the same, provided the velocity is the same. The authors ofk Ref. (15) have used a different expression for the force. This, however, gives only a small difference in the velocity to cause such an
10
VINOGRADOVA AND FEUILLEBOIS
FIG. 6. Ratio of sphere deformation at r = 0 to the gap size for ² = 10−6 , St = 5 in the no-slip limit (solid curve) and for various β0 (dashed curves from top to bottom β0 = 10−4 , 10−3 , 10−2 , 10−1 ): k = −1 (A), 0 (B), 20 (C), ∞ (D).
effect. It was tempting to infer that the difference between our results is related to the fact that, as discussed above, the authors of Ref. (15) lost a factor h 0 /R in their analysis. However, we found that this is responsible only for quantitative difference. We believe there may be two reasons for this discrepancy. First, the authors of Ref. (15) could have a different meaning for the parameter ² (unfortunately, ² is not defined anywhere in their paper), although, in principle, this prefactor should have no qualitative effect on the calculations. Second, we do not exclude their numerical inaccuracy. For completness, the results of our calculations for St = St ∗ and k = 0 are presented in Fig. 7 in the same scale as in (15).
V. FINAL REMARKS
Certain aspects of our work warrant further comments. We are not the first to study the asymptotics of the elastohydrodynamic deformation, but in contrast to the authors of (7, 15) we
restricted ourselves by the calculation of the deformation profile only and did not study the rebound of the particles or the firstorder correction to a force due to deformation. These analytical calculations for the particles allowing slip remain open questions and a big challenge for the theorists. They are beyond the scope of the present paper and will be discussed elsewhere. Below we only specify some of the important details of these problems. We have stressed that because of the behavior of a deformation at short separations the contact of particles is impossible. This conclusion has been made for a situation when deformation is small. It is evident that at some point the deformation asymptotics of the form [4.3] or [4.4] will contradict the original restriction (w ¿ h). Then −∂ H/∂t is no longer equal to the relative velocity of the centers of mass, and the deformation cannot be determined via the pressure profile in the absence of deformation. When deformation is significant (w ∼ h), the rebound of particles becomes possible. This could be only because the elastic strain energy of a deformation can be converted to the kinetic energy of the rebounding particles (5, 6). Such a
11
ELASTOHYDRODYNAMIC COLLISION
FIG. 7. Ratio of sphere deformation at r = 0 to the gap size for the critical Stokes numbers St = St ∗ and k = 0. Dashed curves from top to bottom are the calculation results for β0 = 10−3 , 10−2 , 10−1 , solid curves from top to bottom are the results for the no-slip limit at the same Stokes numbers (St = 6.6180, 4.3331, and 2.1864, respectively).
situation is different from the “after-contact” rebound considered in Ref. (15). We also remark that one can expect that the rebound of the particles allowing slip will be accompanied by the formation of cavities (14, 22, 23). This should lead to some changes (see (24)) in the formulation of a problem. A systematic study of this kind would constitute a significant extension of the pionering work (5). One of the most important relevant problems is the calculation of the correction to pressure (force) caused by deformation. In the general case, it would be incorrect to estimate it, assuming that p(H (r ), t). However, it is straightforward to show that just this hypothesis was the essence of the method suggested in (7). Indeed, these authors considered the change in ˜ H˜ · w(r ), which is the pressure due to deformation to be d P/d first term of the Taylor expansion for pressure in the approximation p(r ) = p( H˜ + w). The application of such an approach was not evident a priori (7), but our results suggest that this can be justified provided the deformation is small. This can be understood from the following simple arguments. Assume that p(r, t) can be presented as p(H (r ), t). If deformation is small, then ∂ H/∂t ∼ −v and Eq. [2.11] can be rewritten in the form µ ¶ H2 1 ∂p H3 − α(k, H ) − bH α(k, H ) −v = Rµ ∂ H 3 2 ¶ µ R ∂w . [5.1] × 1+ · r ∂r
It follows that the approximation p ∼ p(H, t) can be justified only in the case ∂w r ¿ , ∂r R which is exactly the original assumption w ¿ H˜ . The calculation of the correction for pressure for the colliding slippery particles will be analyzed in the forthcoming publications. ACKNOWLEDGMENTS This work was started while O.I.V. was visiting the Laboratory de Physique et M´ecanique des Millieux H´et´erog`enes de l’ESPCI, and we thank E.J. Hinch for helpful discussions during the earlier stage of this study. It was continued at the Institute of Physical Chemistry and completed at the University of Mainz with financial support from the Alexander von Humboldt Foundation. O.I.V. has benefited from valuable conversations with L.R. White.
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VINOGRADOVA AND FEUILLEBOIS
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