Ind. Eng. Chem. Res. 2010, 49, 12461–12470
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Meniscus-Shear Particle Detachment in Foam-Based Cleaning of Silicon Wafers with an Immersion/Withdrawal Cell V. A. Andreev,† J. M. Prausnitz, and C. J. Radke* Chemical Engineering Department, UniVersity of California, Berkeley, California 94720-1462, United States
New experimental data collected at Lam Research Corporation and theoretical analyses are presented for aqueous-foam cleaning of silicon wafers contaminated with strongly adhered 90 nm Si3N4 particles (Freer et al. 2010). We analyze the distribution of contaminant removal along the wafer surface and the influence of foam quality in a vertical rectangular slot upon wafer immersion/withdrawal. At zero foam quality, particle removal along the wafer surface is uniform. Increased foam quality leads to improved overall removal. Removal, however, is no longer uniform with larger detachment rates toward the bottom of the wafer. To explain the observed nonuniform particle removal, we adopt a binary-collision model that demands a linear dependence of removal rate on the surface shear rate. Perturbation analysis provides the distribution of the wall shear rate along the wafer surface in an unfoamed solution. Calculations show that the wall shear rate on the wafer surface is strongly peaked in the meniscus just above the liquid-filled slot. Thus, with no foam present, removal in the meniscus zone dominates the overall removal process. Because the time of exposure to this high shear is the same for all parts of the surface, we obtain uniform cleaning. With foam bubbles present, the wall shear rate in the slot is enhanced, leading to significant removal in the bulk of the slot. Because the residence time of a wafer in the bulk cleaning solution varies for different parts of the wafer, contaminant removal in the bulk of the slot depends on the vertical position. Combined particle removal in the meniscus zone and in the slot leads to the observed nonuniform distribution of contaminant particles remaining on the wafer surface. Increasing foam quality increases the slot wall shear rate and, hence, the removal rate inside the immersion/ withdrawal cell. 1. Introduction Scrupulous wafer cleanliness is required in the semiconductor industry, especially as feature sizes diminish. Current manufacturing of microelectronic chips can require several hundred processing steps, each susceptible to contamination.1 Consequently, there is a need for improved cleaning technologies. A successful cleaning procedure must effectively remove particulate contamination from a wafer surface but not damage existing structures. Many successful cleaning procedures have been developed, including undercutting,2 brush scrubbing,3-5 megasonics,6-9 cryogenic aerosols,10,11 and fluid jets.12 However, these procedures can induce damage for small feature sizes. In a previous paper,13 we described a new aqueous, foambased technology for silicon-wafer cleaning developed at Lam Research Corporation. In this method, a foamed aqueous suspension flows tangentially in the laminar regime past a wafer surface. Suspended solids in the continuous aqueous phase are essential for the removal of adhered particles. Detachment kinetics is well described by binary collision between a suspended surfactant solid and an adhered contaminant particle. In the presence of foam, removal efficiencies of submicrometer silicon-nitride contaminants can approach 100% after a few immersion/withdrawal events.13 Significant advantages follow from slow flow plus minimal chemical needs due to the large volume of gas in the cleaning solution. We report here results of new experiments performed at Lam Research in the previously described immersion/withdrawal cell13 where, for different foam qualities, the wafer is repeatedly inserted and withdrawn at a constant rate from a vertical, * To whom correspondence should be addressed. Tel.: +1-510-6425204. Fax: +1-510-642-4778. E-mail:
[email protected]. † Currently at PATENTICA LLP, 15 Malaya Morskaya St., St. Petersburg, Russia 190000.
constant-gap slot containing the cleaning solution. According to the collision mechanism, shear is essential for particle removal. In the slot geometry, those particles adhered near the bottom of the wafer experience more cleaning time relative to those particles located near the top of the wafer. Hence, more contaminant should be removed near the wafer bottom compared to that near the wafer top. However, we observe that for unfoamed cleaning suspensions, particle removal is nearly uniform along the vertical axis of the wafer surface. This observation cannot be explained by our original collision-kinetic model.13 To improve our theoretical description, we examine the shear distribution along the wafer surface in the immersion/withdrawal dip cell. As the wafer is withdrawn from the slot, a thin liquid film is deposited.14,15 We propose that during wafer immersion and withdrawal, shear in the narrow meniscus region at the commencement of the thin film dominates particle removal. Therefore, we augment the original binary-collision model to account for the shear distribution in the meniscus of the submerging/emerging wafer surface using an extension of Landau-Levich theory.14,16,17 For foamed cleaning suspensions, particle-removal kinetics increases vertically toward the bottom of the wafer surface. The explanation is shear-induced removal in the narrow meniscus region combined with enhanced shear removal in the body of the slot cell due to the presence of foam bubbles. 2. Experiment Details of the materials and experimental procedures have been presented previously.13 Here, we provide only a brief summary. A 200-mm-diameter, unprocessed silicon wafer with a nascent oxygen coating served as the substrate surface. Aqueous-suspended silicon-nitride (Si3N4) contaminant par-
10.1021/ie1012954 2010 American Chemical Society Published on Web 10/27/2010
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Figure 1. Immersion/withdrawal dip cell for 200 mm wafers. (a) Side view of the wafer surface contaminated with Si3N4 particles. (b) Top view of apparatus. (c) Slots in the tank guide the wafer edges and maintain a constant gap between the wafer surface and the tank wall. The wafer is immersed and withdrawn in the 9 mm gap.
ticles were deposited by spin coating. Particle loading of the wafers was sparse, between 5 × 103 and 5 × 104 light-point defect counts per wafer, which corresponds to 25 and 2.5 mm2 per particle. The equivalent spherical diameter of the Si3N4 particles was measured in the range 90-900 nm using surface light scattering. The wet-cleaning solution, a proprietary chemical formulation of Lam Research Corporation,18 is an aqueous suspension of saponified ammonium-based fatty-acid solids. In an alkaline aqueous solution, insoluble fatty-acid solids are negatively charged; they exhibit a platelet morphology.13,19 The mean size of the solids is much larger than that of the adhered particles by over a factor of 100. Specific fatty acids, solution alkalinity, and solids concentrations and size ranges may be found in ref 18. In addition, the continuous aqueous phase also contains a water-soluble surfactant. Some of the cleaning solutions also contained foamed gas bubbles. Foam was generated using point mixing of the cleaning solution with N2. The foam is polydisperse with bubble sizes ranging from 50 to 400 µm. Minimal foam coalescence was observed during the contaminant-removal experiments. Figure 1 shows the immersion/withdrawal or dip cell. Wafers were immersed and withdrawn sequentially by hand from a 13 × 185 × 220 mm open vertical slot with guides on either side and filled with the cleaning solution. The wafer dipping speed was 50 mm/s for 4 s each for immersion and subsequent withdrawal. After each withdrawal, the wafer was rinsed with distilled/deionized (DI) water and dried using an OnTrak Synergy DSS200 rinser/drier. The number of adhered particles on the wafer was then measured using a KLA-Tencor SP1 Classic counter and compared to the number before cleaning. These data determine the particle-removal efficiency, defined as the fraction of particles removed from the wafer after cleaning. A single wafer was processed four times in the dip cell by repeated immersion/withdrawal events with intervening rinsing, drying, and subsequent quantification of the adhered contaminant population. No effect on particle removal was observed for static dwell time in the dip cell or for rinsing time.13
Figure 2. Contaminant-particle removal as a function of vertical position, x, along the wafer relative to the wafer bottom for different qualities, Q, of foam and for time corresponding to four passes of the immersion/withdrawal sequence. Cleaning solution contains a 0.5 wt % solids suspension. Theory lines correspond to model parameters: k1γ ) 0.56, k2γ ) 0.15, R1 ) 0.88. Figure 8 displays the resulting best-fit foam parameter β for each foam quality.
3. Results Figure 2 shows the fraction of contaminant particles A remaining on the wafer surface, ΓA(t)/ΓA(0), as a function of centerline vertical position x relative to the wafer bottom, where ΓA(t) denotes the number of particles adhered per unit area after a fixed number of immersion/withdrawal events corresponding to cleaning time, t. In this figure, zero on the abscissa corresponds to the wafer bottom (x ) 0), while positive x positions increase toward the wafer top. Results are shown for foam qualities, Q, ranging from zero to 90%. Cleaning time in Figure 2 corresponds to nd ) 4 immersion/withdrawal events (i.e., a total cleaning time of 32 s). Solids concentration in the cleaning solution is 0.5 wt %. Similar results were found for solids concentrations ranging from 0.5 to 4 wt %. At the higher solids concentrations, foam has little influence because removal efficiencies approach 100%.13 No difference in contaminantparticle counts was found at vertical slices taken at various horizontal locations across the wafer. Lines in Figure 2 correspond to binary-collision theory described later. For the unfoamed suspension in Figure 2 (i.e., Q ) 0), the fraction of remaining particles is about 0.55 at the bottom and middle parts of the wafer but decreases slightly near the top where the wafer was manually held by a PEEK tweezer. For Q ) 0.2, the fraction of particles remaining is significantly lower, about 0.25, and nearly uniform along the wafer surface. Particle removal increases with foam quality. Starting with foam quality Q g 0.4, however, we observe a gradual increase in particle concentration toward the wafer top (i.e., close to vertical positions of 150 mm and larger). For Q ) 0.625, the fraction of particles remaining is close to that for Q ) 0.75. The dependence of contaminant-particle concentration on vertical position changes from slightly decreasing at Q ) 0 to significantly increasing at Q ) 0.4 and higher. Particle removal near the top of the wafer is independent of foam quality. Because cleaning time under shear is largest near the bottom of the wafer, results for Q < 0.4 were unexpected, as was the shift to larger particle removal near the wafer top at foam qualities larger than 0.4.
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constant in eq 1 can be expressed in terms of a collision crosssection σAB, velocity of solids with respect to the wafer surface uB, and collision efficiency ξ: k0j ) ξjuBσAB
Fˆ , j ) 1, 2 Fˆ BVB
(2)
where Fˆ B is the mass density of solids B, Fˆ is the suspension mass density, and VB is the size-average volume of a surfactantsolid platelet. For small distances from the wafer surface, z, the fluid velocity directly adjacent to the substrate surface is u ) γ˙ z, where γ˙ is the wall shear rate. Upon assuming that the velocity of a surfactant solid is equal to the velocity of the carrier fluid calculated at the center of the platelet solid, it follows that uB ) γ˙ z. Further, for a solids-particle collision to occur, the z coordinate of the solid center must be less than or equal to hB/2 + hA (assuming that the solid is oriented parallel to the surface), where hB is the thickness of the solid and hA is the size of a contaminant particle. Accordingly, eq 2 becomes Figure 3. Logarithm of contaminant particle removal as a function of vertical position along the wafer relative to the wafer bottom for different qualities of foam after nd ) 4 passes through the immersion/withdrawal sequence. Cleaning solution contains a 0.5 wt % solids suspension. Lines are empirical least-squares linear fits. Table 1. Results of Least-Square Linear Fit of Retention Data ln(ΓA(t)/ΓA(0)) versus Vertical Position x Q
slope × 103, mm-1
0 0.2 0.4 0.625 0.75 0.9
-0.45 ( 0.08 -0.05 ( 0.16 1.37 ( 0.12 3.39 ( 0.18 4.07 ( 0.22 7.11 ( 0.36
To emphasize removal behavior, Figure 3 replots the data of Figure 2 on semilogarithmic coordinates. Retention data are well represented by an exponential dependence on wafer vertical position. Table 1 lists the best-fit slopes of linear fits to the data. The slopes of the straight lines in Figure 3 for Q ) 0 and Q ) 0.2 are small and negative, indicating uniform removal within experimental uncertainties. Starting from Q ) 0.4, however, all slopes are clearly positive, increasing with Q. 4. Theory In our previous paper,13 we established that without foam, particle removal is a first-order process in the surface concentration of adhered contaminant particles and first-order in the volume concentration of solids in the cleaning solution. Because contaminant particles are polydisperse in size and shape,13 the overall cleaning kinetics are a combination of simultaneous firstorder detachment kinetics with a distribution of rate constants.20 To a good approximation, the distribution of the rate constants can be represented as bimodal.13 Consequently, for a constant concentration of solids in the aqueous solution, the removal fraction as a function of cleaning time, t, obeys a double exponential decay:13 ΓA(t)/ΓA(0) ) R1 exp(-k01CBt) + R2 exp(-k02CBt)
(1)
where ΓA is the surface density of contaminant particles, k10 and k20 are second-order rate constants in the bimodal distribution, R1 and R2 ) 1 - R1 are weighting fractions of the bimodal distribution corresponding to the two cleaning rate constants, and CB is the weight fraction of solids in the aqueous dispersion. Analogous to chemical-reaction kinetic theory,13 each rate
k0j ) ξj(hB /2 + hA)γ˙ σAB
Fˆ ) kjγγ˙ , j ) 1, 2 Fˆ BVB
(3)
Equation 3 provides the theoretical basis for explaining the particle-retention profiles in Figures 2 and 3. The effect of shear rate on cleaning kinetics is now explicit. To understand the dip-cell cleaning kinetics in Figures 2 and 3, we consider the distribution of residence time and shear rate along the wafer surface. If shear along the dip cell were uniform, then, according to eqs 1 and 3, particle retention at a given vertical location is determined only by the residence time of the wafer in the cleaning solution. As illustrated in the schematic of Figure 4, residence time is linearly proportional to decreasing vertical position along the wafer surface. It is zero at the top of the wafer and equal to total immersion/withdrawal time at the bottom. Consequently, during a dip cycle, particle retention is unchanged at the top of the wafer and exponentially decreases from top to bottom. This theoretical result, however, does not agree with all experimental results in Figures 2 and 3. In particular, for the cleaning solutions without foam (Q ) 0), Figures 2 and 3 show that particle removal is significant and everywhere uniform along the wafer surface. For cleaning solutions with foam (Q > 0.2), removal is better at the wafer bottom but is also significant at the top. These observations, along with eq 3, suggest that wall shear is not uniformly distributed in the dip cell. Toward an improved theory, we hypothesize that for unfoamed suspensions where vertical removal is uniform, cleaning takes place only in a narrow, high-shear region of the dip cell. Because this region is narrow, variation of residence time within this zone is negligible. The narrow high-shear region is most likely located in the meniscus where the liquid-gas interface is curved and capillary forces are significant (see Figure 4). 4.1. Meniscus Shear Stress. To quantify this hypothesis, we compare the calculated wafer shear rate in the meniscus with that in the bulk of the slot. The cleaning suspension without foam is considered an incompressible Newtonian fluid of constant viscosity. We represent the wafer as a translationally invariant vertical plate neglecting edge effects. Likewise, the dip cell is represented as an infinite reservoir of fluid. Any influence of the dip-cell walls is neglected because the distance between the walls and the wafer is sufficiently smaller than the capillary length, lc ) (σ/Fg)1/2 ) 2 mm, where σ is the surface tension (the surface tension of the aqueous surfactant solution is19,21 σ ) 40 mN m-1), F is the density of the fluid, and g
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Ind. Eng. Chem. Res., Vol. 49, No. 24, 2010 Table 2. Numerical Constants in Appendix A
A1 A2 A3 R B1
withdrawal
immersion
1 0 0 -0.125 4.393
0 0.796 0 -0.125 0
B2 B3 C0 C1 C2
withdrawal
immersion
0 0 2.769 0.092 1.338
5.623 3.310 1.204 2.172 1.338
slip limit can be reproduced without considering surfactant diffusion, if instead of zero tangential stress at the interface, the tangential velocity of the interface is set equal to the velocity of the plate.28 We expect that the behavior of the cleaning fluid is bounded by the limits of surfactant-free and surfactantimmobilized interfaces. Appendix A outlines calculation of the flow field and wall shear rate for these two limits for both immersion and withdrawal events. Asymptotic analysis gives the following expression for the wall shear rate in the system γ˙ )
Figure 4. Side view of the dip cell. The wafer is immersed into and withdrawn from the dip cell containing cleaning fluid at a constant speed, U. Vertical scale is in millimeters. x˜ is the vertical coordinate in the reference frame of the cell. x is the vertical coordinate in the reference frame of the wafer, and dw is the wafer diameter.
represents gravity. Modeling of the first immersion is difficult because the wafer is initially dry, leading to a moving contact line and a nonintegrable stress singularity.22 However, second and subsequent immersions do not involve a dynamic contact line because the wafer is covered by a liquid film deposited during the previous withdrawal. As described below, the limiting film thickness is set by the deposition process of the withdrawal stage. Here, we consider only second and subsequent immersion and withdrawal events. The problem of coating of a semi-infinite plate withdrawn at a constant velocity from a liquid bath was solved by Landau and Levich.14,16,17 In the limit of low capillary numbers (here Ca ≈ 1.3 × 10-3), a formal solution is available by matched asymptotic expansion.23-27 Unfortunately, shear stress in the meniscus region has not been evaluated. In addition, plate immersion has not been considered. Most previous solutions of the Landau-Levich problem focus on a surfactant-free gas-liquid interface where the tangential stress on the interface vanishes. Park studied the effect of an insoluble surfactant on the thickness of the deposited film during withdrawal.25 He found that for fast surface diffusion of the surfactant, a low withdrawal speed, and weak variation of the surface tension with surfactant concentration, the deposited film thickness approaches its value for a surfactant-free interface. In the opposite limit of a surfactant-immobilized interface (i.e., 2 no-slip), the deposited film is 4 /3 times thicker than that of a surfactant-free interface, and the interface velocity approaches that of the plane. Intermediate film thickness and surface velocity are bracketed by these two limits. All results of the second no-
σ1/6U1/3F1/2g1/2 (1) 1/3 [λ1F(0)(X) + λ-1 + O(Ca2/3)] 1 F (X) Ca 2/3 η (4)
where Ca ) ηU/σ is the capillary number, U is the velocity of immersion or withdrawal, η is the fluid viscosity, λ1 ) 21/2/C2 for a surfactant-free interface while λ1 ) 21/6/C2 for a surfactantimmobilized interface, and C2 ) 1.338 is a numerical constant. F(0)(X) and F(1)(X) are universal functions different for immersion and withdrawal and are determined in Appendix A, where X is the dimensionless vertical coordinate (the X axis points downward). Equation 4 describes the distribution of wall shear rate in the reference frame fixed to the cell. The dimensionless coordinate X is related to the dimensional x˜ coordinate in Figure 4 by x˜ )
( Fgσ )
1/2
[Ca1/3(λ2hj(0) 0 X - s) - √2]
(5)
where λ2 ) 1, s ) -C1/2, and hj0(0) ) C2/2 for a surfactantfree interface and λ2 ) 4-1/3, s ) -21/6C1, and hj0(0) ) 25/6C2 for a surfactant-immobilized interface. Here, hj(0) 0 corresponds to the leading-order contribution to the thickness of the vertical deposited film. The constant C1 is given in Table 2. Figures 5 and 6 show, respectively, the profiles of the wall shear rate in the system for withdrawal and immersion, calculated using eqs 4 and 5 under the conditions of the dipcell experiment (F ) 103 kg m-3, U ) 5 × 10-2 m s-1, η ) 10-3 Pa s, and σ ) 40 mN m-1). To establish the shear-rate profiles in Figures 5 and 6, calculations must be carried out to O(Ca1/3). At O(1), drainage in the deposited vertical film is negligible,25 and the shear rate is zero. Film drainage emerges at O(Ca1/3),27 and the wall shear rate in the deposited vertical film is no longer zero. However, the O(Ca1/3) term in the perturbation analysis gives only a small contribution to the shear rate in the meniscus zone (less than 10%) and to the depositedfilm thickness (about 1%) for the capillary number considered, Ca ) 1.3 × 10-3. During withdrawal in Figure 5, the wall shear rate increases rapidly starting at 4 mm above the horizontal air-water interface (i.e., at x˜ ) -4 mm) to reach a single maximum of about 1800 s-1 for a surfactant-free interface and 1500 s-1 for an immobilized interface. After the maximum, the shear rate decreases rapidly toward zero in the solution bulk. Figure 5 shows that the width of the high-shear-rate band is about 3 mm, or about 1.5% of the total wafer length. During immersion in Figure 6, the shear rate oscillates: it has both negative and positive values. Oscillations in shear rate increase in magnitude upon approach
Ind. Eng. Chem. Res., Vol. 49, No. 24, 2010
Figure 5. Variation of the wall shear rate along the wafer surface during withdrawal as calculated from second-order perturbation theory for surfactant-free and surfactant-immobilized air/water surfaces. Zero on the abscissa axis corresponds to the flat horizontal liquid surface. The physical parameters are σ ) 40 mN m-1, F ) 103 kg m-3, U ) 5 × 10-2 m s-1, and η ) 10-3 Pa s. The parameters δ1 and δ2 mark the interval of integration in eq 12 for the rate-constant calculation.
Figure 6. Variation of the wall shear rate along the wafer surface during immersion as calculated from second-order perturbation theory for surfactant-free and surfactant-immobilized air/water surfaces. Zero on the abscissa axis corresponds to the flat horizontal liquid surface. The physical parameters are surface tension σ ) 40 mN m-1, density F ) 103 kg m-3, velocity U ) 5 × 10-2 m s-1, and viscosity η ) 10-3 Pa s. Parameters δ1 and δ2 mark the interval of integration in eq 12 for calculation of the rate constants.
to the flat liquid-gas interface, reaching a maximum of 3800 s-1 for a surfactant-free interface and 3100 s-1 for an immobilized interface located at approximately 2.7 mm from the interface. This maximum is followed by a negative shear region with a minimum at -2.4 mm; thereafter, the shear rate increases, approaching zero in the bulk. The width of the high-shear-rate zone during immersion is also about 3 mm. Negative values of the wall shear rate in Figure 6 indicate that the x˜ component of fluid velocity decreases from the wafer surface, as illustrated in Figure 7 where h˜(x˜) is the film-thickness profile in the stationary cell reference frame. Because velocity in eq 2 is measured with respect to the wafer surface, a negative wall shear rate corresponds to the motion of solids against the x˜ axis (the x˜ component of fluid velocity is negative), while positive shear rate corresponds to motion along that axis. Only the magnitude of the solids’ velocity is important for collision
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Figure 7. Shape of the liquid-gas interface during immersion for a surfactant-free interface as determined from O(Ca1/3) perturbation theory. h˜ is the local film thickness. The calculations are performed for the same conditions as those in Figure 6. The velocity field is shown schematically by arrows.
statistics. Hence, we use the magnitude of the wall shear rate in all equations that follow. Wall shear rate in the meniscus must be compared to that in the bulk of the slot and to that in the deposited vertical film. If we assume a simple linear slot velocity profile, then for a 6-mm gap and a plate velocity of 50 mm/s, we obtain a bulk shear rate γ˙ (b) ) 8 s-1. This shear rate is 2 orders of magnitude smaller than that in the meniscus (see Figures 5 and 6). For the conditions of the particle-removal experiments, the shear rate in the film is 217 s-1 for a surfactant-free interface and 274 s-1 for a surfactant-immobilized interface. Thus, the wall shear rate in the deposited film is 1 order of magnitude smaller than that in the meniscus. Consequently, after taking into account the exponential dependence of removal on the shear rate in eqs 1 and 3, we conclude that for unfoamed suspensions, essentially all cleaning takes place in the meniscus region. The small width of the high-shear zone, 1.5% of total wafer length, explains the uniform removal observed for Q ) 0 in Figures 2 and 3. Mathematical solution of the meniscus pull-out and plunge-in hydrodynamics confirms our conjecture that the origin of the uniform particle-removal distribution in unfoamed suspensions is a peaked distribution of shear rate focused in a narrow region of the meniscus above the wafer. 4.2. Kinetic Rate Constants. Previously, we assumed a uniform shear rate in the slot of the dip cell and timeindependent rate constants.13 However, since the wall shear rate changes with position in the dip cell, the wafer surface experiences shear that changes with time as it passes through the cell; therefore, rate constants in eq 3 depend on time. Because the shear-rate history is different for each position along the plate, removal is position-dependent. Thus, to apply the binary-collision model in the varying shear field of the dip cell, we must modify eq 1. We recognize that the rate constants now depend on time for each position x on the wafer surface: 2
ΓA(t, x)/ΓA(0, x) )
∑ R exp(-C ∫ j
j)1
B
ndtd 0 kj
0
dt)
(6)
where nd is the number of dip cycles and td is the time of a single complete dip cycle (immersion plus withdrawal). Since the rate constants k0j depend linearly on shear rate, eqs 3 and 6
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predict removal in different types of cleaning equipment provided that the shear-rate profile is known. We assume that all dips are equivalent and separate the contributions of each dip into immersion and withdrawal events ΓA(nd, x) ) ΓA(0, x)
2
∑ R exp[-C n ( ∫
td/2
j
B d
0
j)1
kjγγ˙ wd(t;x) dt +
∫
td/2
0
kjγγ˙ im(t;x) dt)] (7)
where γ˙ wd is the shear-rate history during withdrawal and γ˙ im is the shear-rate history during immersion. Equation 7 is written in the reference frame of the wafer surface. In this reference frame, the shear rate changes with time. However, Figure 4 illustrates that the shear-rate history experienced by different locations x on the wafer surface is not the same. For example, during immersion, locations near the bottom of the wafer experience meniscus shear and a prolonged shear in the bulk of the slot, whereas locations near the top of the wafer experience meniscus shear and minimal shear time in the slot. Hence, x is a parameter in eq 7. Because the speed of immersion and withdrawal U is constant, time in eq 7 is related to the coordinate x˜ in the reference frame of the dip cell as dt ) dx˜/ U. Corresponding changes in the limits of integration can be established using Figure 4; the x dependence of shear rate now appears only in the limits of integration: ΓA(nd, x) ) ΓA(0, x)
∑ R exp[2
j
j)1
C B nd (U
∫
-x k γ˙ (x˜) dw-x jγ wd
∫
dw-x
-x
dx˜ +
kjγγ˙ im(x˜) dx˜)
]
(8)
where dw is the wafer diameter. Equation 8 gives the fraction of particles remaining at a given position x on the wafer surface after nd immersion/withdrawal events. Fractional particle retention is expressed through the wall-shear-rate history in the dip cell and depends on the position along the wafer surface through the limits of integration. It is convenient to split the integrals in eq 8 into three parts corresponding to the deposited film (f), the meniscus (m), and the bulk of the slot (b), respectively:
∫
dw-x
-x
kjγ(γ˙ wd + γ˙ im) dx˜ )
∫
δ2
δ1
∫
δ1
-x
(m) kjγ(γ˙ wd + γ˙ (m) ˜+ im ) dx
(f) kjγ(γ˙ wd + γ˙ (f) ˜+ im) dx
∫
dw-x
δ2
(b) kjγ(γ˙ wd + γ˙ (b) ˜ im ) dx
(9)
Here, δ1 is the border between the deposited film and the liquid meniscus marked in Figures 5 and 6. Likewise, δ2 is the border between the meniscus and the bulk slot in Figures 5 and 6. Shear rate in the deposited film is independent of x˜ (see Figures 5 and 6), and the variable x varies from 0 to dw ) 200 mm, much lager than either δ1 or δ2. Without error, we set δ1 ) 0 and δ2 ) 0 in the first and the third integrals of eq 9. 4.2.1. Unfoamed Suspensions. For unfoamed suspensions, the wall shear rate in the bulk of the slot is negligible, and the last integral in eq 9 vanishes, giving ΓA(nd, x) ) ΓA(0, x)
∑ R exp[2
j
j)1
CBndkjγ (f) ((γ˙ wd + γ˙ (f) im)x + U
∫
δ2
δ1
for Q ) 0 there is indeed a very shallow decrease of particle concentration with x, as indicated by the negative slope of the straight line for Q ) 0 displayed in Table 1. This variation, however, is very weak, indicating that the shear rate developed in the deposited liquid film, γ˙ (f), is insufficient for collision removal of particles. Therefore, we omit the first term in the exponent of eq 10 to obtain
(m) (γ˙ wd + γ˙ (m) ˜) im ) dx
]
(10)
Equation 10 predicts an exponential decrease of contaminant particle retention with the x position along the wafer surface. Figures 2 and 3 and Table 1 show that in the experimental results
ΓA(nd) ) ΓA(0)
2
∑ R exp[-C n k j
(m) B d j ]
(11)
j)1
where k(m) ) j
kjγ U
∫
δ2
δ1
(m) (γ˙ wd + γ˙ (m) ˜ im ) dx
(12)
Equation 11 describes the dependence of particle concentration on the number of dips and the concentration of solids. It predicts that particle removal is uniform along the wafer surface in agreement with the experimental results in Figures 2 and 3. Removal occurs only in the narrow meniscus region. By using the pre-exponential factors R1 ) 0.88 and R2 ) 0.12, determined in our previous work,13 and by then fitting eq 11 to experimental removal data for Q ) 0 in Figure 2, we find k1(m) ) 39.6 and k2(m) ) 10.8. The resulting solid line for Q ) 0 is shown in Figure 2. To obtain rate constants k1γ and k2γ, which are independent of the specific flow field, we must evaluate the integral in eq 12. We define δ1 and δ2 in eq 12 by shear rates exceeding 50% of the shear rate in the deposited film. For withdrawal, the interval of integration is [δ1 ) -3.45, δ2 ) -1.75 mm] for a surfactant-free interface and [δ1 ) -3.71, δ2 ) -1.55 mm] for a surfactant-immobilized interface. For immersion, we have [δ1 ) -2.98, δ2 ) -1.63 mm] and [δ1 ) -3.02, δ2 ) -1.30 mm], respectively. Following numerical evaluation of the wallshear-rate quadrature in eq 12, Figures 5 and 6 give 3210 mm/s for a surfactant-free air/water interface and 3900 mm/s for a surfactant-immobilized interface. For the complete dip cycle, we take a simple average of these two integrals, 3600 mm/s. Using this approximation, the experimental rate constants of found above and U ) 50 mm/s, we find from eq 12 that k1γ k(m) j ) 0.56 and k2γ ) 0.15. Since kjγ is universal independent of shear rate, these two rate constants permit quantitative prediction of contaminant removal for different types of cleaning hardware using eqs 3 and 6, provided that the shear-rate history is known. 4.2.2. Foamed Suspensions. We next analyze the effect of foam on cleaning. Figures 2 and 3 show that fractional removal for all foam qualities converges to approximately the same value of 0.5 at the wafer top, x ) 200 mm (the abrupt decrease in concentration at x ) 200 mm is a result of holding by the tweezer; this effect is reproduced for all wafers). As opposed to unfoamed suspensions, contaminant removal for foamed suspensions changes with position along the wafer surface. Only the meniscus effect can contribute to removal in the top wafer region. Since foam bubbles are large, O(100 µm), we neglect the possible presence of bubbles in the thin deposited film. Accordingly, foam provides little to no enhancement to removal in the meniscus region. Removal in Figures 2 and 3 is better at the wafer bottom, indicating that with foam present, the contribution from cleaning in the slot of the dip cell is important. In this case, the wall shear rate,γ˙ , is enhanced in the wetting liquid films between the wall and adjacent-most foam bubbles. Rate constants kjγ are also likely to increase in the presence of foam because of solids confinement and orientation in the liquid film separating foam bubbles from the wall.13 Hence, according
Ind. Eng. Chem. Res., Vol. 49, No. 24, 2010
Figure 8. Variation of foam parameter β with foam quality as determined by fitting eq 13 to the experimental data in Figure 2. The line is a leastsquares fit for Q < 0.9 with slope kQ ) 657.
to eq 3, the removal rate constants in the bulk of the slot are increased and contaminant removal is enhanced. The analysis presented in the previous section remains valid for cleaning with foam up to eq 9. In addition to the second meniscus term on the right of eq 9, we must also retain the last term that describes removal in the bulk of the dip cell. By assuming that the wall shear rate in the bulk of the slot is independent of x˜, we obtain after integration
Figure 9. Overall contaminant particle retention as a function of foam quality after the time corresponding to four immersion/withdrawal events. Cleaning solution contains a 0.5 wt % solids suspension. The solid line is calculated using eq 14 with linear dependence of the rate constants on foam quality and kQ ) 657; the dotted line shows the earlier result.13
Parameter kQ ) 657 was found from a least-squares linear fit to the points in Figure 8. To obtain the overall contaminant removal from the wafer, we integrate eq 13 with respect to x: ΓA(nd) 1 ) ΓA(0) dw
∫
dw
0
ΓA(nd, x) dx ) ΓA(0, x)
2
ΓA(nd, x) ) ΓA(0, x)
∑ R exp[-C n k
2
∑
Rj exp[-CBnd(k(m) j
+
k0j td(1
- x/dw))]
j
j)1
j)1
(13) where td ) 2dw/U. Equation 13 gives the fraction of particles remaining at a given position x on the wafer surface after nd immersions/withdrawals in a foamed suspension. The first term in the exponent of eq 13 accounts for foam-independent cleaning in the meniscus. At the wafer top, x ) dw, meniscus removal is the only contribution to cleaning. The second term quantifies the decrease in adhered particle concentration with position x in the bulk of the slot. As x increases, the last term in the exponent decreases, resulting in reduced contaminant removal. To determine the dependence of the kinetic parameters k0j on foam quality, we assume that foam quality equally influences both removal rate constants k0j ; we introduce foam parameter β defined by k0j ) βkjγ where β varies only with foam quality. Lines in Figure 2 give the best fit to eq 13, yielding β for each foam quality. No improvement in agreement to data was found if the parameters k10 and k20 were fitted independently. Figure 8 shows with filled circles the resulting dependence of parameter β on foam quality Q. All points in Figure 8, except for Q ) 0.9, are well described by a linear function of foam quality, β ) kQQ, where kQ is the foam-influence parameter. In general, the parameter kQ, may depend on the wall shear rate in the system. The linear dependence of β on Q for low and intermediate foam qualities agrees with our previous result,13 where we found that a good description of overall contaminant removal with foam is achieved when the rate constants in eq 11 are multiplied by a linear function of foam quality Q. In this work, we find for qualities less than Q ) 0.9 that the same approach can be used to describe locally the influence of foam quality on removal at a given position along the wafer surface.
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(m) B d j ]
1 - exp[-CBk0j tdnd] CBk0j tdnd
(14)
Equation 14 describes the influence of dip number on the overall removal of contaminant particles from the wafer surface. This result predicts the influence of foam quality using the linear relationship between k0j and Q, k0j ) kjγkQQ, where kQ ) 657. Results for overall contaminant removal are shown as a solid line in Figure 9. Our theory well describes the effect of foam quality on contaminant retention. The dotted curve in Figure 9 corresponds to our previous model.13 Removal results from the two models are close. 5. Discussion Figures 2 and 3 show that for unfoamed and low-quality foamed suspensions, cleaning of Si3N4 contaminant particles from a silicon wafer is uniform along the wafer surface. This result is not consistent with our previously proposed binarycollision mechanism for contaminant removal in a dip cell, as particles near the bottom of the wafer are subject to shear longer than those near the top. The earlier binary-collision analysis assumed that removal occurs at constant shear in the slot of the dip cell.13 For unfoamed suspensions, however, we find here that the shear rate in the meniscus dominates that in the slot. Our hydrodynamic calculations show that the high-shear-rate region in the meniscus adjacent to the wafer is very narrow, with shear rates orders of magnitude larger than those in the slot of the dip cell or in the deposited film. Our calculations indicate that cleaning takes place only in a narrow region of the meniscus, leading to uniform wafer cleaning when no foam is present. Using the newly calculated shear-rate distribution in the meniscus of the dip cell for unfoamed suspensions and
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experimentally determined rate constants13 k1(m) and k2(m), we calculate the shear-rate-independent constants k1γ and k2γ. These two rate constants and the weighting fractions R1 and R2 from the bimodal distribution corresponding to the two cleaning rate constants are then employed in eq 13 to calculate the Q ) 0 curve in Figure 2. The model correctly predicts a uniform distribution of contaminant concentration along the wafer surface after four dips, in good agreement with experimental results. Figure 2 shows that foam has two effects on cleaning: first, foam enhances contaminant removal significantly, and second, it produces nonuniformity of cleaning along the wafer surface. As discussed previously,13 foam cleaning enhancement can arise from three effects. First, foam bubbles orient and confine the platelet solids closer to the wafer surface, increasing the nearsurface concentration of solids and, hence, the collision frequency. Second, foam bubbles increase the near-surface shear rate. If we consider the problem of a moving bubble in a tube,26,29 similar to the plate drag-out problem considered here, we find that the wall shear rates at the front and rear ends of a flowing bubble are significantly higher than that in parabolic pipe flow. We anticipate that analogous enhancement of the wall shear rate takes place in the Plateau borders adjacent to the wafer surface in the foam of the dip cell. Recently, Saugey et al.30 studied wall slip in foams using numerical simulation. For small capillary numbers, they found that the shear rate is negligible in the deposited thin film between the wall and the bubble most adjacent to the wall, whereas in the front and back of each bubble, shear is significantly enhanced. Third, bubbles orient the platelet solids more parallel to the wafer surface. A parallel orientation is likely to be more efficient for particle removal.13 Because all three effects are stronger with an increasing gas fraction, we observe enhanced cleaning with foam. For foamed suspensions, contaminant removal is no longer meniscusdominated. Both the meniscus and the bulk-slot regions contribute to particle removal. The cleaning contribution from the bulk region increases as the foam quality rises. By fitting eq 13 to data on retention profiles, we determine the dependence of the rate constants k0j on foam quality. This dependence is linear for Q < 0.9 but deviates from it significantly for Q ) 0.9, as shown in Figure 8. The model predicts a doubleexponential increase of contaminant concentration with the position on the wafer for all foam qualities. For high foam qualities, the variation of concentration with position is well represented by the double-exponential function. However, for low foam qualities, the experimental distribution of particle retention is not exponential, but rather close to uniform, with retention slowly increasing with position. This discrepancy can be explained by the underlying assumptions used to derive eq 13. First, we assumed that the rate constants in the bulk of the dip cell are the same for all positions. However, this assumption may not hold for low quality foams. At low gas fractions, the bubbles may cream, producing a higher shear rate near the top of the wafer. By a force balance on a bubble accounting for drag and buoyancy, we find that the steady-state velocity of bubbles in the systems is 2.2 × 106a2 m/s, where a is the bubble radius. In our experiments, bubble size ranges from 50 to 400 µm, giving an increase in velocity ranging from 5 to 350 mm/ s. Because the wafer diameter is 200 mm, segregation of the bubbles during the time of the experiment is possible. The resulting nonuniformity of shear distribution in the bulk of the dip cell can compensate for the increase in residence time toward the wafer bottom. Interplay between these opposite trends may result in a relative uniform particulate removal at low Q.
Second, we assumed that foam bubbles have no influence on cleaning inside the meniscus. Foam bubbles may enter the high-shear-rate zone of the meniscus and enhance removal there. Concomitantly, the contribution of the bulk foam to the cleaning may remain negligible at low foam qualities. By integrating eq 13 over the wafer surface, we successfully quantify the overall retention of contaminant particles. Parameters k0j determined from the linear relation with foam quality predict the overall retention of contaminant particles, as demonstrated in Figure 9. Lack of agreement with experimental results for Q ) 0.9 is due to the assumed linear relation between k0j and Q for high foam qualities. By recognizing the important of meniscus shear, the extended binary-collision model successfully represents the nonuniform contaminant removal observed in dip-cell experiments. A theoretical basis for the binary-collision model is presented elsewhere.31 6. Conclusions We analyze new experimental data for the removal of small Si3N4 particles adhered to silicon wafers using a cleaning solution consisting of a foamed aqueous suspension of insolublesurfactant platelet solids.13,18 For an unfoamed solids suspension in an immersion/withdrawal cell, contaminant removal is uniform along the wafer. Introduction of gas bubbles leads to significantly better but nonuniform cleaning. We explain these results by the distribution of the wall shear rate along the wafer surface. Our calculations show that in unfoamed suspensions, the wall shear rate is concentrated in the narrow meniscus region. Therefore, cleaning occurs only in this narrow region, thus producing uniform particle removal. However, introduction of foam bubbles increases cleaning in the bulk of the cell slot due to enhanced shear rate. We improve our previous binarycollision model to take into account two sequential steps of the process: cleaning with foam in the bulk slot and cleaning in the meniscus. The observed nonuniform removal in foamed suspensions is explained by variation of the residence time with position on the wafer surface. Our improved model gives a good representation of the experimental removal data for different foam qualities and positions, supporting the fundamental binarycollision mechanism. Acknowledgment V.A.A. was partially supported by an unrestricted gift from Lam Research Corporation. Nomenclature a ) radius of foam bubbles (m) Ca ) capillary number CB ) weight fraction of solids B dw ) diameter of wafer (m) F(0)(X) ) O(1) contribution to the dimensionless wall shear rate F(1)(X) ) O(Ca1/3) contribution to the dimensionless wall shear rate g ) gravitational acceleration (m s-2) hB ) thickness of platelet solid (m) hA ) size of contaminant particle (m) hj(0) 0 ) O(1) contribution to the thickness of the vertical deposited film kj0 ) rate constant, independent of CB (s-1) kjγ ) rate constant, independent of γ˙ and CB (s-1) kQ ) foam influence factor kj(m) ) rate constant defined by eq 12 of the text
Ind. Eng. Chem. Res., Vol. 49, No. 24, 2010 lc ) capillary length (m) nd ) number of immersion/withdrawal events Q ) foam quality (volume fraction of gas) s ) numerical constant defined after eq 5 t ) time of cleaning (s) td ) time of immersion/withdrawal cycle (s) U ) speed of wafer immersion/withdrawal (m s-1) u ) velocity of fluid (m s-1) uB ) velocity of solids B (m s-1) VB ) the size-average volume of solids B (m3) X ) dimensionless vertical coordinate in the reference frame of the cell x ) vertical coordinate in the reference frame of the wafer (m) x˜ ) vertical coordinate in the reference frame of the cell (m) z ) normal distance from the wafer surface (m) Greek Symbols Rj ) weighting fraction corresponding to the cleaning rate constant kj0 β ) coefficient describing the influence of foam ΓA(t,x) ) surface density of contaminant particles A at time t and position x (m-2) γ˙ ) wall shear rate (s-1) γ˙ wd ) wall shear rate during withdrawal (s-1) γ˙ im ) wall shear rate during immersion (s-1) γ˙ (f) ) wall shear in deposited vertical film (s-1) γ˙ (m) ) wall shear in meniscus (s-1) γ˙ (b) ) wall shear in bulk (s-1) δ1 ) x˜ coordinate of the border between the deposited film and the meniscus (m) δ2 ) x˜ coordinate of the border between the meniscus and the bulk slot (m) η ) viscosity of the solution (Pa s) λ1) numerical constant defined after eq 4 λ2 ) numerical constant defined after eq 5 ξ ) collision efficiency σ ) surface tension of the solution (N m-1) σAB ) collision cross-section (m2) F ) density of the solution (kg m-3) Fˆ ) suspension mass density (kg m-3) Fˆ B ) mass density of solids B (kg m-3)
Appendix A. Calculation of Wall Shear Rate for Wafer Immersion/Withdrawal
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and
H(1) XXX ) -
(H(1)(6H(0) - 9) + H(0)((3R + (H(0))3 - 1) (H(0))4 (A4)
Upper signs in eqs A3 and A4 correspond to withdrawal and the bottom signs to immersion; H(0) ) hj(0)/hj0(0). Likewise, H(1) ) hj(1)/(hj0(0))3 and R ) hj0(1)/(hj0(0))3 for a surfactant-free interface, and H(1) ) 4hj(1)/(hj0(0))3 and R ) 4hj0(1)/(hj0(0))3 for a surfactantimmobilized interface. Symbols hj(0) and hj(1) are coefficients in the asymptotic expansion in the transition region:26 hj(xj) ) hj(0)(xj) + hj(1)(xj)Ca1/3 + O(Ca2/3), where hj(xj) is related to dimensional j(1) film thickness h˜ as h˜ ) lcCa2/3hj.25 hj(0) 0 and h0 are limiting values of hj(0)(xj) and hj(1)(xj) as jx f -∞. Equations A3 and A4 were solved in MATLAB using a RungeKutta (4,5) solver in the interval -10 e X e 25 with relative tolerance 10-3. To initiate the integration at X ) -10, we used the asymptotic solution of eqs A3 and A4 as X f -∞ or
Figure 10. Variation of O(1) and O(Ca1/3) contributions to the dimensionless wall shear rate along the wafer surface during withdrawal as calculated from perturbation theory.
The drag-out problem was first successfully solved by Landau and Levich14,16,17 and later revisited using the method of matched asymptotic expansions.23-27 In this appendix, we give the final expressions needed to calculate the wall shear rate and describe the computational procedure. Details of the solution can be found in the literature.14,16,17,25-27,29 Functions F(0)(X) and F(1)(X) in eq 4 of the text are defined by (0) F(0) ) H(0) XXXH
(A1)
(1) (0) F(1) ) H(0) + (H(1) XXXH XXX + 1)H
(A2)
and
where functions H(0)(X) and H(1)(X) are found by solving the following system of ordinary differential equations H(0) XXX ) (
3(H(0) - 1) (H(0))3
(A3)
Figure 11. Variation of O(1) and O(Ca1/3) contributions to the dimensionless wall shear rate along the wafer surface during immersion as calculated from perturbation theory.
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1 H(0) ) 1 + A1 exp((31/3X) + A2 exp - 31/3X × 2 √3 1/3 3 (X + A3) sin 2
(
(
)
)
(A5)
and 1 H(1) ) R + B1 exp((31/3X) + B2 exp - 31/3X × 2 √3 1/3 3 (X + B3) sin 2
(
(
)
)
(A6)
where coefficients A1, A2, A3, R, B1, B2, and B3 are determined from the matching principle; they are listed in Table 2. The resulting functions F(0)(X) and F(1)(X) are shown in Figures 10 and 11 for immersion and withdrawal, respectively. The limiting form of the solution of eq A3 as X f ∞ is 1 H(0) ) C2X2 + C1X + C0 2
(A7)
Coefficients C2, C1, and C0 in eqs 4 and 5 specify the solution completely. These coefficients, determined from the solution of eq A3 at X ) 25, are also listed in Table 2. Once F(0)(X) and F(1)(X) are known, the wall shear rate is calculated using eq 4, where the coordinate X is expressed in terms of x˜ using eq 5. Literature Cited (1) Kern, W. Handbook of Semiconductor Wafer Cleaning Technology; Noyes Publications: Park Ridge, NJ, 1993. (2) Vos, R.; Lux, M.; Xu, K.; Fyen, W.; Kenens, C.; Conard, T.; Mertens, P.; Heyns, M.; Hatcher, Z.; Hoffman, M. Removal of Submicrometer Particles from Silicon Wafer Surfaces Using HF-Based Cleaning Mixtures. J. Electrochem. Soc. 2001, 148, G683–G691. (3) Philipossian, A.; Mustapha, L. Tribological Attributes of Post-CMP Brush Scrubbing. J. Electrochem. Soc. 2004, 151, G456–G460. (4) Xu, K. Nano-sized particles: Quantification and remoVal by brush scrubbing cleaning. Ph.D. Thesis, Katholieke Universiteit, Belgium, 2004. (5) Xu, K.; Vos, R.; Vereecke, G.; Doumen, G.; Fyen, W.; Mertens, P. W.; Heyns, M. M.; Vinckier, C.; Fransaer, J. Particle adhesion and removal mechanisms during brush scrubber cleaning. J. Vac. Sci. Technol. B 2004, 22, 2844–2852. (6) Bakhtari, K.; Guldiken, R. O.; Makaram, P.; Busnaina, A. A.; Park, J.-G. Experimental and numerical investigation of nanoparticle removal using acoustic streaming and the effect of time. J. Electrochem. Soc. 2006, 153, G846–G850. (7) Burkman, D. C.; Deal, D.; Grant, D. C.; Peterson, C. A. Aqueous Cleaning Processes. Handbook of Semiconductor Wafer Cleaning Technology; Noyes Publications: Park Ridge, NJ, 1993; p 139. (8) Qi, Q.; Brereton, G. J. Mechanisms of removal of micron-sized particles by high-frequency ultrasonic waves. IEEE Trans. UFFC 1995, 42, 619–629. (9) Gale, G. W.; Busnaina, A. A. Removal of Particulate Contaminants Using Ultrasonics And Megasonics: A Review. Part. Sci. Technol. 1995, 13, 197–211.
(10) Banerjee, S.; Campbell, A. Photomask and Next-Generation Lithography Mask Technology XII (Proceedings Volume); SPIE Publications: Bellingham, WA, 2005; pp 90-99. (11) Toscano, C.; Ahmadi, G. Particle Removal Mechanisms in Cryogenic Surface Cleaning. J. Adhes. 2003, 79, 175–201. (12) Hirano, H.; Sato, K.; Osaka, T.; Kuniyasu, H.; Hattori, T. DamageFree Ultradiluted HF/Nitrogen Jet Spray Cleaning for Particle Removal with Minimal Silicon and Oxide Loss. Electrochem. Solid-State Lett. 2006, 9, G62–G65. (13) Andreev, V. A.; Freer, E. M.; de Larios, J. M.; Prausnitz, J. M.; Radke, C. J. Silicon-Wafer Cleaning with Aqueous Surfactant-Stabilized Gas/Solids Suspensions. J. Electrochem. Soc. 2010, In Press, doi: 10.1149/ 1.3503572. (14) Levich, V. G. Physicochemical hydrodynamics; Prentice-Hall: Englewood Cliffs, NJ, 1962. (15) de Gennes, P.-G.; Brochard-Wyart, F.; Quere, D. Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, WaVes; Springer: New York, 2004. (16) Landau, L. D. Collected papers of L.D. Landau; Pergamon Press: Elmsford, NY, 1965. (17) Landau, L. D.; Levich, V. G. Dragging of a liquid by a moving plate. Acta Physicochim. URSS 1942, 17, 42–54. (18) Freer, E. M.; de Larios, J. M.; Mikhaylichenko, K.; Ravkin, M.; Korolik, M.; Redeker, F. C. Cleaning compound and method and system for using the cleaning compound. Patent US 7696141, 2010. (19) Kralchevsky, P. A.; Danov, K. D.; Pishmanova, C. I.; Kralchevska, S. D.; Christov, N. C.; Ananthapadmanabhan, K. P.; Lips, A. Effect of the Precipitation of Neutral-Soap, Acid-Soap, and Alkanoic Acid Crystallites on the Bulk pH and Surface Tension of Soap Solutions. Langmuir 2007, 23, 3538–3553. (20) Hutchinson, P.; Luss, D. Lumping of mixtures with many parallel first order reactions. Chem. Eng. J. 1970, 1, 129–136. (21) Averko-Antonovich, I. Y.; Ziganshina, L. R.; Rakhmatullina, A. P.; Akhmed’yanova, R. A. Surface activity of fatty acid salts in aqueous solutions. Rus. J. Appl. Chem. 2004, 77, 595–598. (22) Huh, C.; Scriven, L. E. Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J. Colloid Interface Sci. 1971, 35, 85– 101. (23) Bush, A. W. Perturbation methods for engineers and scientists; CRC Press: Boca Raton, FL, 1992. (24) Kevorkian, J.; Cole, J. D. Multiple scale and singular perturbation methods; Springer: New York, 1996. (25) Park, C.-W. Effects of insoluble surfactants on dip coating. J. Colloid Interface Sci. 1991, 146, 382–394. (26) Park, C.-W.; Homsy, G. M. Two-phase displacement in Hele Shaw cells: theory. J. Fluid Mech. 1984, 139, 291–308. (27) Wilson, S. D. R. The Drag-out Problem in Film Coating Theory. J. Eng. Math. 1982, 16, 209–221. (28) de Gennes, P. G. Deposition of Langmuir-Blodgett layers. Colloid Polym. Sci. 1986, 264, 463–465. (29) Bretherton, F. P. The motion of long bubbles in tubes. J. Fluid Mech. 1961, 10, 166–188. (30) Saugey, A.; Drenckhan, W.; Weaire, D. Wall slip of bubbles in foams. Phys. Fluids 2006, 18, 053101-12. (31) Andreev, V. A.; Prausnitz, J. M.; Radke, C. J. Surface collision theory for suspension-based cleaning of silicon wafers. Submitted to J. Appl. Phys. 2010.
ReceiVed for reView June 16, 2010 ReVised manuscript receiVed September 22, 2010 Accepted October 5, 2010 IE1012954
