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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 <1%). This is because the renormalization of slip length is negligible for b ≪ R and the residue boundary curvature eﬀect described by f 2 is also very small for small h. So for this case (b ≪ R, h ≪ R), the overall boundary curvature eﬀect is not important and Vinogradova’s result is reliable. However, as h increases, the diﬀerence between f * and f v increases monotonously and already becomes signiﬁcant around h = 87

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renormalization of the slip length is not important due to b ≪ R. However, our full result, taking into account the residue boundary curvature eﬀects via f 2, is slightly but observably smaller than f1/h or f v/h. As we further increase the slip length to b = 500, the three curves are distinguishable from one another, indicating that both the renormalization of slip length (replacing b by b̃) and the boundary curvature eﬀect described by f 2 are important. Because the resistance force quickly decreases as the gap width is increased, the quantitative diﬀerences are more easily observed at small h. For the largest slip length studied in this paper, b = 5000, we ﬁnd f */h is only slightly smaller than f1/h because the contribution due to f 2 becomes much smaller than that due to f1. However, Vinogradova’s original result, without renormalizing the bare slip length, signiﬁcantly overestimates the hydrodynamic resistance force. Therefore, it is clear that, in general, the boundary curvature eﬀects on hydrodynamic forces should not be discarded, especially when the slip length is comparable to the particle size. We end this section by brieﬂy discussing the consequence of our results for some important problems in colloid science and ﬂuid dynamics. First, our results may play a critical role in the process of particle coagulation. Similar to Vinogradova’s results, our results predict the possibility of particle collision in a ﬁnite time in the absence of additional attractive surface forces, although this is not allowed in the Reynolds theory due to the divergence of hydrodynamic resistance as particles get close. As observed from Figure 3d, when particle surface is coated with very hydrophobic material so that the slip length is comparable to the particle size, the hydrodynamic resistance at short distance can be greatly reduced by the boundary curvature eﬀects. At h = 1, it is nearly half of that predicted by Vinogradova’s original work, and this should make the particle coagulation easier. Our theory revealed the importance of boundary curvature eﬀects in determining the particle collision time. However, we will leave more detailed and quantitative analysis on the coagulation process to future studies. Second, our results may have important consequences on the dynamic measurements of hydrophobic forces using the drainage technique.7 In the typical experimental setting,8 optical interferometry is used to determine the separation between two crossed cylinders, and one of the cylinders is mounted on a cantilever spring, through which the total restoring force Fk is applied. The simple force balance gives Fk = Fs + Fh, where Fh is the hydrodynamic force and Fs is the total surface force. Both Fk and Fs depend on the distance h, while Fh also depends on the approaching velocity of the two cylinder particles. In general, Fs can have several contributions from rather diﬀerent mechanisms,9 depending on physical/chemical properties of the surfaces and the solvent medium. The contribution of repulsive forces (such as solvation forces and double-layer forces) is diﬃcult to be determined and often avoided/neglected. The remaining attractive contributions to Fs include the DLVO van der Waals force that is inversely proportional to h2, the real hydrophobic force usually characterized by a double exponential function of h and possibly an extra power-law hydrophobic attraction. Because only the hydrodynamic force is supposed to depend on the velocity, it can be clearly identiﬁed and subtracted from Fk to obtain the total surface force. Then, because of their distinct distance dependence, diﬀerent contributions to Fs such as the DLVO van der Waals force and the hydrophobic force can be

identiﬁed. Clearly, the accurate evaluation of the hydrodynamic force is central to the above scheme to measure the surface forces with the drainage technique. It is thus important to include the boundary curvature eﬀects, according to what we have done in this paper. The qualitative and quantitative consequences of boundary curvature eﬀects on surface force measurements should be carefully examined in diﬀerent experimental settings. Third, in recent years, there have been many attempts to measure the slip lengths for various solid−ﬂuid interfaces. An important class of experiments10,11 is based on Vinogradova’s theory. In a typical setting, the viscous force induced by a sphere moving orthogonally to a plane has been employed, using either a bead glued on an atomic force microscopy cantilever or a surface force apparatus. Then, the slip length can be determined by accurate measurements of the hydrodynamic force and of the gap width h (and the rate of change of h, or the relative sphere-plane velocity), based on the analytic expression for the hydrodynamic resistance force. Although in this work our analytic results are only obtained for two identical spheres, it should be not diﬃcult to generalize them to the sphere-plane geometry. For such a geometry, we expect that the hydrodynamic force can also be written as a sum of two contributions, similar to eqs 23−25. The ﬁrst part will be identical to Vinogradova’s corresponding result, but the bare slip lengths should be replaced by the renormalized ones, similar to what we write in eq 24. The second part will represent the extra boundary curvature eﬀects. A detailed analysis of the boundary curvature eﬀects in slip length measurements will be presented elsewhere. However, it is already clear from our results that the boundary curvature eﬀects can signiﬁcantly reduce the hydrodynamic force and can play a critical role in correctly determining the true (bare) slip length.

■

CONCLUSIONS Vinogradova’s work on drainage of thin ﬁlm between two hydrophobic spheres represents a signiﬁcant improvement over traditional Reynolds theory of hydrodynamic lubrication. Her work shows that ﬂuid slippage over solid surfaces can have important consequences on the hydrodynamic interactions, both qualitatively and quantitatively. We have further improved her work by taking into account the boundary curvature eﬀects, which are found to be especially important when the slip length is comparable to the sphere radii. For two identical spheres, we have obtained analytic expressions for the hydrodynamic resistance force, distinguished from Vinogradova’s result by two aspects: ﬁrst, the bare slip length should be replaced by the curvature-renormalized eﬀective slip length; second, an extra contribution should be included to take care of the residue boundary curvature eﬀects. The derived correction on Vinogradova’s result can play a dramatic role in the coagulation processes of colloidal particles. Furthermore, such correction should be taken into account for experimental investigations on surfaces forces or slip lengths with the drainage technique.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]; [email protected] Notes

The authors declare no competing ﬁnancial interest. 88

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ACKNOWLEDGMENTS The research grant to Yongli Mi by the China 1000 Talent Scholar Grant is greatly appreciated for the support of this work.

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REFERENCES

(1) Vinogradova, O. I. Drainage of a Thin Liquid Film Confined between Hydrophobic Surfaces. Langmuir 1995, 11, 2213−2220. (2) Lauga. E.; Brenner,M. P.; Stone, H. A. Microﬂuidics: The No-Slip Boundary Condition. arXiv preprint cond-mat/0501557, 2005. (3) Neto, C.; Evans, D. R.; Bonaccurso, E.; Butt, H. J.; Craig, V. S. Boundary Slip in Newtonian Liquids: A Review of Experimental Studies. Rep. Prog. Phys. 2005, 68, 2859−2897. (4) Bocquet, L.; Barrat, J.-L. Flow Boundary Conditions from Nanoto Micro-Scales. Soft Matter 2007, 3, 685−693. (5) Einzel, D.; Panzer, P.; Liu, M. Boundary Condition for Fluid Flow: Curved or Rough Surfaces. Phys. Rev. Lett. 1990, 64, 2269− 2272. (6) Potanin, A. A.; Uriev, N. B.; Muller, V. M. Criterium of the coagulation of particles under dynamic conditions while taking into account the slipping of liquid at the surface of particles. Kolloidn. Zh. 1988, 50, 493−499. (7) Vinogradova, O. I. Implications of Hydrophobic Slippage for the Dynamic Measurements of Hydrophobic forces. Langmuir 1998, 14, 2827−2837. (8) Chan, D. Y. C.; Horn, R. G. The Drainage of Thin Liquid Films between Solid Surfaces. J. Chem. Phys. 1985, 83, 5311−5324. (9) Israelachvili, J. N. Intermolecular and Surface Forces, 3rd ed.; Academic Press: Burlington, MA, 2011. (10) Cottin-Bizonne, C.; Cross, N.; Steinberger, A.; Charlaix, E. Boundary Slip on Smooth Hydrophobic Surfaces: Intrinsic Effects and Possible Artifacts. Phys. Rev. Lett. 2005, 94, 056102. (11) Zhu, L.; Attard, P.; Neto, C. Reconciling Slip Measurements in Symmetric and Asymmetric Systems. Langmuir 2012, 28, 7768−7774.

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