Article pubs.acs.org/Langmuir
Drainage of a Thin Liquid Film between Hydrophobic Spheres: Boundary Curvature Effects Angbo Fang*,† and Yongli Mi*,†,‡ †
Department of Chemistry, Tongji University, 1239 Siping Road, Shanghai, P. R. China 200092 Department of Chemical and Biomolecular Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
‡
ABSTRACT: We investigate theoretically the drainage of a thin liquid film confined between two hydrophobic spheres. Such a problem has been considered in Vinogradova’s seminal work, emphasizing the role of slippage. However, it does not include the boundary curvature effects, which may become especially important when the slip lengths are comparable to the sphere radii. We present a reformulation of the hydrodynamic boundary conditions, with the boundary curvature effects fully taken into account. It is shown that such effects not only renormalize the effective slip lengths but also give new contributions to the pressure and drag force, neglected so far. We focus on the symmetric case of two identical spheres with the same radii and slip lengths and obtain analytical expressions for the pressure as well as the drag force. The boundary curvature corrections to the drag force are analyzed and found to be more important for more hydrophobic spheres. The implications of our results are discussed for the coagulation processes of colloids and measurements of surface forces or slip lengths with the drainage technique.
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INTRODUCTION Hydrophobization of a solid surface is widely used in many industrial processes. Quantitatively calculating the surface forces between hydrophobic bodies immersed in a fluid is crucial for understanding such fundamental phenomena as adhesion, wetting, film stability, cavitation, and coagulation. The drainage process of a thin liquid film confined between two hydrophobic surfaces is fundamentally important because it not only plays a key role in the coagulation process of colloids but also can provide useful information on surface properties by measuring the drainage rate. The seminal theoretical calculation by Vinogradova1 on the drainage of a thin liquid film between two hydrophobic spheres has received considerable attentions. By employing Navier’s partial slip boundary conditions for the liquid flow near hydrophobic surfaces, she generalized Reynolds theory for hydrodynamic lubrication to obtain analytic expressions of the pressure and drag force for arbitrary values of slip lengths as well as for arbitrary radii of the approaching spheres. Her results has been widely cited and employed in experimental interpretations on measuring the hydrodynamic forces or slip lengths. However, by revisiting her work, we found some limitations (or some undeclared approximations) in her derivation. Most importantly, Vinogradova neglected the curvature effects in the hydrodynamic boundary conditions, and this may make her results unreliable in case the slip lengths are comparable to the sphere radii. While traditionally colloidal particles are usually on the micrometer scale and the slip lengths for most solid surfaces are on the nanometer scale, it is not rare when the two length scales are comparable: on one hand, superhydrophobic surfaces with very large slip lengths (up to micrometer) have been © 2013 American Chemical Society
fabricated and employed in various contexts; on the other hand, nanoparticles are frequently encountered/utilized in the fields of colloidal science or microfluidics/nanofluidics. The boundary curvature effects should be generally relevant in studying the properties of colloidal particles, especially the coagulation process. Even if the effects are weak for a large particle with a small slip length, they should be included when there is a need for accurately evaluating the hydrodynamic forces in a range of distance, for example, in the dynamic measurements of hydrophobic forces. We reformulate the theory of thin liquid film drainage between hydrophobic spheres to fully account for the boundary curvature effects. For the two spheres being of the same radius and with equal slip lengths on their surfaces, analytical solutions are obtained for the pressure and drag force, which includes additional contributions missed in Vinogradova’s work.
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SLIP BOUNDARY CONDITIONS ON CURVED SURFACES For a long time, the no-slip boundary conditions were considered reliable to supplement the Navier−Stokes equation to describe the dynamics of confined liquid flow. However, in the recent two decades, this has been questioned by more and more researchers from both theoretical perspectives and experimental measurements.2 It is now generally accepted that hydrodynamic slippage occurs at a wide variety of solid− liquid interfaces.3 The importance of accounting for slippage has been firmly established for flow over hydrophobic surfaces, Received: October 24, 2013 Revised: November 27, 2013 Published: December 10, 2013 83
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DRAINAGE OF LIQUID BETWEEN TWO HYDROPHOBIC SPHERES Consider two spherical undeformable particles S1 and S2 with radii R1 and R2, respectively, immersed in a Newtonian liquid (Figure 1). The distance between particles, h, is assumed to be
especially for liquids confined by a channel of a width comparable to the slip length. The slip length, b, defined by the extrapolation distance within the solid at which the fluid velocity becomes zero (relative to the solid wall), also has a definite mechanical/thermodynamic meaning.4 Assume v is the liquid flow velocity adjacent to the solid wall and vw is the velocity of the wall. Let the superscripts “⊥” and “∥” stand for the projection of a vector along the direction normal and tangential to the solid surface, respectively. We have v⊥ = v⊥w if the liquid cannot penetrate into the solid. Thus the velocity slip, defined by vs ≡ v − vw = v∥ − v∥w, is in a plane tangential to the solid wall. We denote by n the unit vector in the axis normal to the surface, pointing into the liquid. For a liquid element at the liquid−solid interface, there are two types of forces acting on it, and they should be balanced because the interfacial inertia effect can usually be neglected. With η as the liquid viscosity, −(η/b)vs is the friction force exerted by the wall on a unit area of adjacent liquid. Assuming the liquid flow stress tensor is σ adjacent to the wall, the tangential force per unit area exerted by the liquid side is the surface traction n·σ projected into the tangential plane. The boundary condition along the tangential axis is given by the following force-balance requirement: (n ·σ ) ·(I − nn) −
η s v =0 b
Figure 1. Schematic representation of the drainage of a thin liquid film between two spheres with radii R1 and R2, respectively. The distance of closest approach between the two surfaces is h, and the velocities of the upper and lower spheres are, respectively, given by −V/2 and V/2. The appropriate cylindrical coordinate system, (z,r), is also shown.
(1)
small compared with either R1 or R2. The particles approach each other along the line connecting their centers. To explore the benefits of symmetry, we employ a cylindrical system (z,r) of coordinates, with the axis z coinciding with the line connecting the sphere centers. We put the origin of coordinates in the midpoint of the two sphere centers. Furthermore, we work in a moving inertia frame so that the velocities of the upper and lower spheres are, respectively, given by −V/2 and V/2. In the inner region that is close to the origin of coordinates, the surface of sphere S1 may be described locally as a paraboloid of revolution:
For a planar surface, when a rectangular coordinate system is used for both in the bulk and at the boundaries, eq 1 reduces to the following form:
(n ·∇)v =
vs b
(2)
This is the familiar form of the partial slip BC used by most authors. However, we should keep in mind that eq 2 applies only to planar surfaces. For generally curved surfaces, we should refer to eq 1 as the correct formulation of the partial slip BC. At the first step, we would better represent σ in the local orthogonal coordinate system defined by n and v∥. Then, we reexpress eq 1 in the global orthogonal coordinate system often used for flow far away from the boundaries. With nk and vk (k = 1, 2, 3) as the components of n and v∥ in this global coordinate system, eq 1 can be manipulated to become
z=
v =0 b
z=− (3)
(5)
h 1 r2 − + O(r 4) 2 2 R2
(6)
The steady-state flow of liquid in the gap between particles should satisfy the following equations:1 (1) The continuity equation
We note that the second term is contributed by curvature effects and vanishes for planar surfaces. For a spherical surface with the radius of curvature R, this term can be rewritten as v∥/ R. Thus we can reduce the slip BC to the form for a planar surface:
v (n ·∇)v − l = 0 b*
h 1 r2 + + O(r 4) 2 2 R1
In a similar way, the surface of sphere S2 may be characterized by
s
(n ·∇)v + vk∇ nk −
Article
∂vz 1 ∂ + (rvr ) = 0 ∂z r ∂r
(7)
where vz and vr are the projections of liquid flow velocity on the axes z and r, respectively. (2) The r-component hydrodynamic equation
(4)
⎡ ∂ 2v ⎛ ∂v ∂v ⎞ v ⎤ ∂p 1 ∂ ⎛ ∂vr ⎞ ⎜r ⎟ − r2 ⎥ + μ⎢ 2r + ρ⎜vz r + vr r ⎟ = − ⎝ ∂z ∂r ⎠ ∂r r ∂r ⎝ ∂r ⎠ r ⎦ ⎣ ∂z
with b replaced by the renormalized slip length, b*, defined through (1/b*) = (1/b) − (1/R). The importance of such curvature-induced renormalization of the slip length was noticed a long time ago.5 With the establishment of the correct formulation of the slip BC on a spherical surface, we are now ready to attack the problem of liquid drainage between two hydrophobic spheres.
(8)
where ρ is the liquid density, p is the pressure, and μ is the dynamic viscosity. (3) The z-component hydrodynamic equation 84
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⎡ ∂ 2v ⎛ ∂v ∂v ⎞ ∂p 1 ∂ ⎛ ∂vz ⎞⎤ ⎜r ⎟⎥ ρ⎜vz z + vr z ⎟ = − + μ⎢ 2z + ⎝ ∂z ∂r ⎠ ∂z r ∂r ⎝ ∂r ⎠⎦ ⎣ ∂z
assumption used in Reynolds theory of hydrodynamic lubrication: the flow velocity gradient along the z axis dominates over the flow velocity gradient along the r axis. This assumption, along with the incompressibility condition, can dramatically simplify the hydrodynamic equations. Namely, eq 8 will reduce to
(9)
The previous three equations are to be amended by appropriate boundary conditions to fully determine the fluid dynamics. Near a hydrophobic surface, a finite slip boundary condition should be applied tangentially, and a nonpenetrability condition should be applied normally. Assume the slip lengths for the surfaces of S1 and S2 to be b1 and b2, respectively. The boundary conditions are given by vz −
μ
(12)
(13)
We remark that the same essential assumption has been applied to the expression of the tangential stress under our slip boundary conditions. While eq 13 implies p is a function of only r, integrating eq 12 leads to
4
at z = (h/2) + (1/2)(r /R1) + O(r ) and rvr v ⎤ V ⎡ ∂v 1⎡ r ⎤ = ; ⎢ r + r ⎥ − ⎢vr + V⎥ = 0 R2 2 ⎣ ∂z R 2 ⎦ b2 ⎣ 2R 2 ⎦ (11)
vr = −
at z = −(h/2) − (1/2)(r /R1) + O(r ). Importantly, the flow nonpenetrability and flow slip boundary conditions should be applied to the locally normal and tangential directions, respectively. Therefore, the boundary curvature effect should be carefully incorporated into the expressions for the flow-wall relative velocity projected along normal and tangential directions as well as for the locally tangential fluid stress. We emphasize again that in the literature the tangential flow slip boundary conditions are often formulated as the proportionality of the flow velocity slip and the flow velocity gradient along the normal direction. Nevertheless, such a formulation is only valid for planar surfaces, as we made clear in the preceding section. Our general formulation in the preceding section is valid for any curved surface and clearly preserves the thermodynamic/mechanical meanings for the boundary conditions along either the normal or tangential directions. Compared with Vinogradova’s seminal work, our formulation is different in several aspects. First, Vinogradova adopted a coordinate transformation, z′ = z + r2/2R2, to simplify the boundary conditions at the surface of S2 (which becomes planar). However, by such a transformation (z′,r) is no longer an orthogonal coordinate system and it should result in much more complicated expressions for the differential operators in the hydrodynamic equations. It is necessary to justify that correction terms due to the nonorthogonality of the coordinate system are negligible. We keep the orthogonal cylindrical coordinate system and here eqs 7−9 are exactly valid. With the origin located on the midpoint of the two spheres and the velocities of the spheres are given by ± V/2, the boundary conditions are also formulated in a symmetric way. This can further simplify the derivation. Second, Vinogradova included the curvature-induced correction in her formulation of boundary conditions along the locally normal direction near the surface of S1, but she apparently discarded it in subsequent calculations. Last, but not least, the curvature effects on the tangential slip boundary conditions are completely neglected in her formulation. This can be very important when the slip length is comparable to the corresponding radius of curvature. Now we proceed to explore quantitative consequences of our improved formulation. With the gap width h, a small parameter compared with the sphere radii, we can still apply the essential 2
∂p ∂r
∂p ∼0 ∂z
(10)
vz +
∂z
∼
2
and eq 9 will reduce to
⎡ ∂v rvr v ⎤ V 1⎡ r ⎤ V⎥ = 0 = − ; − ⎢ r − r ⎥ − ⎢vr − R1 2 R1 ⎦ b1 ⎣ 2R1 ⎦ ⎣ ∂z 2
∂ 2vr
4
1 ∂p 2 z + C1z + C2 2μ ∂r
(14)
where C1 and C2 are two constants to be determined by the two tangential boundary conditions. The general expression for C1 and C2 are rather lengthy and we do not present them here. In the following calculations, we focus on the case of two identical spheres, that is, R1 = R2 = R and b1 = b2 = b, where expressions can be greatly simplified. Namely, we have C1 =
r V 2R H + b ̃
(15)
and C2 = −
1 ∂p H(H + 2b)̃ H 2μ ∂r
(16)
where H ≡ h/2 + (r2/2R) and 1/b̃ = 1/b − 1/R represents curvature-induced renormalization of the slip length. The bare slip length, b, is chosen to be smaller than R throughout this paper so that b̃ > 0. Now, by integrating the continuity eq 7 over z from −H to H and utilizing the normal boundary conditions, we obtain the following differential equation for the pressure ⎡ r H ∂ ⎡ ∂p ⎤ ⎢⎣Xr ⎥⎦ = μ⎢⎣1 − RH+ ∂r ∂r
⎤ ⎥Vr ̃ b⎦
(17)
where X = −2H2(H + 3b̃)/3. Integrating eq 17 twice and noticing that dp/dr = 0 at r = 0 (due to symmetry) and p = 0 at r → ∞ (at the inner boundary of outer region), we arrive at the following expression for the pressure: p ≡ p1 − p2
(18)
where p1 (r ) =
3μV R ⎡ H ⎛ 3b ̃ ⎞⎤ ⎢1 − ln⎜1 + ⎟⎥ 4H 3b ̃ ⎣ H ⎠⎦ 3b ̃ ⎝
(19)
and 85
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Figure 2. Comparison of the correction functions f * and f v as functions of the gap width. Four different slip lengths are considered: (a) b = 5, (b) b = 50, (c) b = 500, and (d) b = 5000. The sphere radii are set to be R = 10 000, and a significant range of gap width h = 1−1000 is plotted.
p2 (r ) =
3μV H ⎡⎛ h ⎞ ̃ h ⎜1 + ⎟g (b ) − g (0) ⎢ H 6b ̃ ⎣⎝ 2b ̃ ⎠ 3b ̃ ⎤ ⎛ h ⎞ ̃⎥ ⎟g (3b ) − ⎜1 + ̃ ⎝ ⎠ ⎦ 6b
that obtained by Vinogradova, due to the fact that ∫ R0 p1r dr ≫ ∫∞ R p1r dr. Therefore, we have Fz ≡ (20)
∫r
∞
dr
1 R ≡2 cot−1 H+y h + 2y
r R(h + 2y)
f1 =
∫0
≈
∫0
(21)
We note that both p1 and p2 are positive. Furthermore, except for the fact that Vinogradova got an erroneous global negative sign, p1 is identical to Vinogradova’s corresponding expression for the pressure (noting that our definitions of both H and R differ from her corresponding definitions by a factor of 2), except that here b is replaced by b̃. The contribution of p2, completely missing in Vinogradova’s work, arises here due to the boundary curvature effects. It tends to reduce the value of pressure everywhere in the inner region. We emphasize that either our expression or Vinogradova’s for the pressure is valid only for the inner region. The hydrodynamic resistance force acting on sphere S1 is given by Fz = −
∫0
R
(23)
where f* ≡ f1 − f 2
with g defined by g (y ) =
3πR2μV f* 2h
⎛ dv ⎞ ⎜ −p + 2μ z ⎟2πr dr ⎝ dz ⎠
R
p1 2πr dr
∞
= (2)
p1 2πr dr ⎤ h ⎡⎛ h ⎞ ⎛ 6b ̃ ⎞ ⎟ ln⎜1 + ⎢⎜1 + ⎟ − 1⎥ h ⎠ 6b ̃ ⎣⎝ 6b ̃ ⎠ ⎝ ⎦
(24)
and f2 = =
∫0
R
p2 2πr dr
⎛ R ⎞⎤ ⎛R⎞ h R⎡ ⎛ R ⎞ ⎟ − 2f ⎜ ⎟ − f ⎜ ⎟⎥ ⎢3f ⎜ ̃ ̃ ̃ ⎝ ⎠ ⎝ h + 6b ̃ ⎠⎦ h 3b 6b ⎣ ⎝ h + 2b ⎠
(25)
with f (x ) =
(22)
⎤ 1 ⎡π x − (1 + x) arctan x ⎥ ⎦ 2 x 3/2 ⎢ ⎣
(26)
f * defined by eq 23 is often called the correction function to the hydrodynamic resistance force because the Reynolds theory, without considering the slippage effects as well as the boundary curvature effects, predicts that the resistance is simply given by 3πR2μV/2h. The form of f1 is identical to the correction function obtained by Vinogradova1 except that here the slip length b is replaced by the renormalized slip length b̃. We note that f1 (as well as p1) is positive, indicating a resistance force pushing sphere S1 away from sphere S2. The novel
The contribution to the force from the outer region is of smaller order and can be neglected. Furthermore, the force is predominated by the contribution of the pressure in eq 22. It is somehow debated whether the upper limit of integration in eq 22 should be equal to ∞ or the particle radius; Potanin et al.6 used R, but Vinogradova used ∞ as the upper limit of integration, respectively. Nevertheless, for the case of h ≪ R, the force obtained by Potanin et al. is essentially identical to 86
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Figure 3. Comparison of the friction coefficients obtained with and without including boundary curvature effects. Four different slip lengths are considered: (a) b = 5, (b) b = 50, (c) b = 500, and (d) b = 5000. The sphere radii are set to be R = 10 000, and a significant range of gap width h = 1−1000 is plotted. The friction coefficients, computed according to f */h, f v/h, and f1/h are denoted as “full”, “Vinogradova”, and “Vinogradova (rectified)”, respectively.
100. This is due to the fact that f 2 becomes more and more important and is no longer negligible at intermediate gap with (100 < h < 1000). Interestingly, while f v is a monotonously increasing function of h, f * is not monotonous. It first increases, achieves it maximum at around h = 100, and then decreases. In addition, at the largest gap width studied here, h = 1000, f v almost attains its asymptotic value 1, but the value of f * is smaller than it by nearly 20%. The behavior of f * for larger slip lengths is more regular: similar to f v, f * is a monotonously increasing function of h, approaching its asymptotic value at large h. For b = 50, we could still observe the closeness of f v and f * at small h. However, for b = 5000, f * is significantly smaller than f v in the whole range of gap width we studied (1 < h < 1000). The boundary curvature effects on the correction factor are best illustrated for large slip length and large gap width (still much less than the particle size). In this case, Vinogradova’s result is no longer accurate. It is more of interest to study the influence of boundary curvature effects on the friction coefficient, f */h. This quantity presents the part of the hydrodynamic resistance force (Fz), which depends on the film thickness, while the remaining part remains fixed in a particular experiment. In Figure 3, we plot f */h (denoted as “full”) versus f v/h (denoted as “Vinogradova”) as functions of h in the range of 1−1000. For comparison, the friction coefficient f1/h (denoted as “Vinogradova (rectified)”), obtained by replacing the bare slip length by the renormalized slip length in Vinogradova’s expression, is also presented. Again, we consider four cases with different orders of magnitude in the slip length, b = 5, 50, 500, and 5000. For b = 5, it is difficult to distinguish the three curves from one another. The reason is that for such nearly hydrophilic spheres, the overall boundary curvature effect contributes little to the hydrodynamic resistance force. For b = 50, there is little difference between f 1 /h and f v /h, signifying that the
correction term, f 2, arises from the residue boundary curvature effects not captured by simply renormalizing the slip length. Here we cannot replace the upper limit of integration for p2 by ∞; otherwise, it would diverge. From a physical point of view, the correction part of the pressure, p2, is intimately related to the fluid-sphere boundary, and beyond the region r ≤ R we should have p2 = 0. Thus we are consistent in evaluating the resistance force based on eq 22. Furthermore, because p2 is positive, f 2 is also positive. Thus the correction due to f 2 always leads to reduction of the hydrodynamic resistance force. In addition, replacing the bare slip length b by the renormalized slip length b̃ also leads to reduction of f1 because f1 is a monotonously decaying function of the slip length and b < b̃. Therefore, by including the boundary curvature effects of the spheres, we obtain a hydrodynamic resistance force always smaller than the results reported previously without considering such effects.
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DISCUSSION Let us now analyze the results in more detail. First we compare our correction function f * with Vinogradova’s result (denoted by f v), which is essentially given by eq 24 but with b̃ replaced by b. Because our results are valid for h ≪ R, we set R = 10 000 and consider the range of gap width h ≤ 1000. In Figure 2, we show how the correction functions vary as the gap width changes, for four cases with distingushing bare slip lengths: b = 5, 50, 500, and 5000. For b = 5 (Figure 2a), we find that f * agrees well with f v at small gap width (for h < 10 the difference is
