Particle Removal in Linear Shear Flow: Model Prediction and Experimental Validation M. L. Zoeteweij ∗ , J. C. J. van der Donck and R. Versluis TNO Science and Industry, P.O. Box 155, 2600 AD Delft, The Netherlands Received in final form 28 December 2008

Abstract In many industrial processes, particle contamination is becoming a major issue. Particle detachment from surfaces can be detrimental, e.g., during lithographic processing. During cleaning, however, detachment of particles is aimed for. However, until recently, only little was known on the mechanism of particle detachment due to flowing gasses. In high throughput applications, large gas velocities are likely to occur at certain locations in the system. It is important to test particle behavior experimentally under all conditions that may arise. Therefore, the aim of this study is to be able to predict the risk of particle detachment by modeling. For this purpose, particle–surface interaction is studied for micrometer-sized particles. Based on the particle Reynolds number, critical particle diameters were determined for which the flow-induced forces on the particles (drag and lift forces) are larger than the attractive forces between the particle and the surface (van der Waals force). Among the different possible particle motions (lift, sliding and rotation), particle rotation turns out to be the mechanism responsible for particle removal. A critical particle diameter was defined for which attractive and flow-induced forces are equal. Calculated values of the critical particle diameter agree with the experimental results within a few micrometers. This removal mechanism model can thus be used to calculate the cleaning efficiency of a flow, and for determining the probability of unwanted detachment of particles from surfaces in ultra-clean production or processing environments. © Koninklijke Brill NV, Leiden, 2009 Keywords Particles, shear flow, particle–flow interaction, particle–surface interaction, removal, cleaning, resuspension, aerosolisation

1. Introduction For many industrial applications, a clean environment is essential to prevent processing errors. The sticking of particles to the wall surface and the release of particles from the wall surface are a major concern. In this article, particle–surface interaction and the removal and transport of particles by gas flows at atmospheric conditions are studied in detail. Based on the *

To whom correspondence should be addressed. E-mail: Marco.Zo[email protected]

© Koninklijke Brill NV, Leiden, 2009

DOI:10.1163/156856109X411247

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particle and flow properties, the particle behavior can be predicted. The possible types of motion are categorized as: 1. Complete removal by lift. 2. Sliding over the surface. 3. Rolling over the surface. The relevant parameters and expressions are derived. In the current research, only the removal of solid particles with a flow of gas is investigated. The model was verified against experimental data. 2. Forces Acting on a Particle A particle on a surface experiences several forces. These forces can be related to the particle itself, or are induced by the flow field at the location of the particle. The relevant forces acting an a particle are shown schematically in Fig. 1. 2.1. Adhesion Forces The gravity force and the van der Waals interaction are directly related to the particle, and are not affected by the flow. Other attractive forces (e.g., electrostatic and capillary) are not included in the current model. The gravity force FG is given by FG = m · g = ρ · V · g,

(1)

with m the particle mass and g the gravitational acceleration. The mass m equals density ρ times volume V .

Figure 1. Forces and torque acting on a particle in linear shear flow. The relative strengths of the lift (FL ), friction (Ff ), drag (FD ) and adhesion force (Fad ) determine the type of motion that will be induced in the particle with radius r and contact radius a. For rotation, the magnitudes of the torques Cz and r need to be comparable. Reproduced from [1].

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The second contribution is the van der Waals force, which for a spherical particle is given by [2] AH · d 2a 2 , (2) 1 + FvdW = z0 · d 12z02 in which AH is the material dependent Hamaker constant, d the particle diameter, z0 the particle-to-surface distance (usually assumed to be z0 = 0.4 nm) and a the contact radius of the particle, as defined in Fig. 2. The second term within brackets corrects for the contact area increase due to particle deformation. For a non-deformed spherical particle, the contact area is a point so a = 0 and the second term cancels out. For simplicity, only spherical particles on flat surfaces are investigated, and surface roughness of the particle or the surface is not included. The plastic deformation of the particle is considered in the analysis. According to the Johnson–Kendall– Roberts (JKR)-theory, the contact radius of deformation can be calculated from the work of adhesion Wa of the particle [3]. Note that both the particle and the surface can deform, so properties of both materials, indicated by subscripts 1 and 2, respectively, determine the contact radius a via the deformation constant K. 2 AH 4 1 − γ12 1 − γ22 −1 3 3π · Wa · d a= , K= + , (3) , Wa = 2K 3 E1 E2 12π · z02 where E1,2 are the Young’s moduli and γ1,2 the Poisson ratios of the two materials. Based on the expressions for gravity and van der Waals forces, these two contributions can be compared. For particles smaller than 50 µm, the gravity interaction (proportional to d 3 ) can be neglected, and the van der Waals force (proportional to d) is the most relevant interaction. Electrostatic forces are not taken into account in the current analysis, since the experimental system is grounded. The experiments

Figure 2. Spherical particle of size d on a flat surface. The distance of nearest approach is defined by z0 (a). The contact radius a is defined by the deformation of the particle when an external load P is applied, shown in (b). Without external force, P = 0, and only the adhesion force Fad is relevant. Reproduced from http://web2.clarkson.edu/projects/crcd/me437/downloads/5_vanderWaals.pdf (a) http://web2.clarkson.edu/projects/crcd/me437/downloads/6_JKR.pdf (b).

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were carried out in a conditioned environment at 22◦ C and 45% humidity. Results by Cardot et al. [4] show that at this level of relative humidity the contribution of the capillary force to the total adhesion force is strongly reduced. 2.2. Removal Forces Any flow will exert forces on the particles, which can be classified in lift and drag forces. Near a no-slip surface, a boundary layer will be present in the flow. In this boundary layer, there is a gradient in the stream-wise velocity. Since close to the wall the velocities are small, a linear velocity profile is assumed in the boundary layer. This assumption is validated by calculating the thickness of the viscous sublayer. The velocity gradient results in a lift force FL acting perpendicular to the stream-wise direction, pointing towards the region with higher velocities. Since the particles are in the boundary layer of the fluid flow, these particles will experience a lift force FL expressed by [5] 1/2 2 ρ ∂u Up . (4) FL = 1.615 · η · d η ∂y The (∂u/∂y) term denotes the gradient in y-direction of the velocity profile u(y). The term must be evaluated at the centre of the particle, at y = d/2. The subscript p indicates that the flow velocity Up at the particle location should be used to calculate the lift force. When the lift force is larger than the sum of the attractive forces, the particle will be lifted from the surface. The drag force represents the force that is exerted on a body by a flow in streamwise direction. The drag force is given by 1 (5) FD = ρU 2 · CD · A, 2 with ρ the fluid viscosity, U the characteristic fluid velocity at the position of the body (=Up ), CD the drag coefficient and A the frontal area of the particle. The value of the drag coefficient CD for small particles (particle Reynolds number in the Stokes range 10−4 < Rep < 2) is given by CD = 1.7009 · f = 1.7009

24 . Rep

(6)

The term f = 24/Rep represents the friction on a spherical particle falling in a static fluid column [6]. The pre-factor 1.7009 corrects for the effect of the wall, which changes the flow pattern around the particle and thus the drag force [7]. The static friction of a body on a surface is expressed by the static friction factor μs . The minimum force needed to overcome static friction is given by Ffriction = μs F⊥ . The net force F⊥ perpendicular to the surface is given by the difference between attractive components (gravity force and van der Waals force) and the lift force pointing in opposite directions. In situations where the drag force is larger than the static friction force, the particle will start sliding over the surface.

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The third type of motion that the particle may experience as a result of flow interaction is rotation, which is related to the moment of surface stresses, MD [8] as 1 MD = ρU 2 · CM · V . (7) 2 Here V is the volume of the particle and CM a constant which equals CM = 0.943993 · f = 0.94339

24 . Rep

(8)

The factor 0.943993 corrects for the presence of the wall [5, 7]. The expression f = 24/Rep is similar to that used in the drag force calculations. Substitution of CM in the equation for MD results in MD = 0.943993 · η · 2π · d 2 · Up .

(9)

2.3. Force and Torque Balance Figure 1 shows which forces and torques act on the particle. The balance between these forces and torques determine by which of the three possible mechanisms the particle will be removed. The particle will be elevated from the surface when the lift force is larger than the attractive van der Waals force and the gravity force, i.e., FL FvdW + FG . When neglecting gravity (which is a valid assumption for particle diameters of 50 µm or smaller) and setting the lift force larger or equal to the van der Waals force, the critical particle diameter for lift can be calculated as AH dlift √ 3/2 12z02 1.615 ρη( ∂u 2 ∂y ) −3/4 −3/4 (AH )1/2 ∂u −1/4 ∂u = (ρη) = ζlift . (10) 3.11 · z0 ∂y ∂y The density ρ and dynamic viscosity η are fluid properties and the value (∂u/∂y) represents the local shear rate in the flow at the center of the particle. The critical particle diameter for lift shows a −3/4 power relation with the local shear rate with a constant ζlift , which includes particle and fluid properties. For particle sliding, the inequality reads FD μs (FV + FG + FL ) = μs (FV + FG − FL ) with the lift force acting oppositely to the other two contributions. For small particles, the gravity can again be neglected and the critical particle diameter for sliding can be calculated as −1 ∂u μs · AH μs · AH −1 ∂u −1 = η = ζsliding . (11) dsliding 2 1.7009·3π ∂u 2 ∂y ∂y 96.2 · z0 12z0 η ∂y 2 The critical diameter for sliding is inversely proportional to the local shear rate in the flow. The particle material and fluid properties are included in the constant ζsliding . Particle rotation will occur when MD + FD L1 + FL L2 (FvdW +

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FG )L2 . L2 equals the contact radius a. Assuming that the contact radius is very small compared to the particle diameter, the value of L1 can be approximated by the particle radius L1 = d/2. MD is the moment of the surface stresses. Since all terms have different dependences on particle diameter d, a single expression cannot be derived for rotation. 3. Particle Reynolds Number The behavior of a fluid flow can be characterized by the dimensionless Reynolds number U L U Lρ = , (12) Re = ν η with U and L the characteristic velocity and length scale in the flow, respectively. The kinematic viscosity (ν) equals the dynamic viscosity (η) over the fluid density (ρ). In a similar way, a particle Reynolds number Rep = Up dρ/η can be defined which describes particle behavior in a flow. The particle diameter is indicated by d, and Up indicates the characteristic stream-wise velocity at the location of the particle centre. The balances shown in the previous section can be related to the particle Reynolds number Rep . For each motion, a critical particle Reynolds number can be defined based on the properties of the flow and the fluid [8]. The gravity and van der Waals forces are combined and indicated by Fattraction in the expressions. When the particle Reynolds number is larger than one of the critical Reynolds numbers defined below, the particle will start moving accordingly. ρ Fattraction , (13) Relift = 2 1.615d ρ ∂u η η

Resliding = Rerotation =

∂y

μs · Fattraction 1.7009 · 3π + μs · 1.615 · d

ρ η

Fattraction · L2 0.94399 · 2π · d + 1.7009 · 3π · L1 + 1.615 · d

∂u ∂y

ρ , η2

ρ η

ρ . ∂u η2 L 2 ∂y

(14)

(15)

Based on the particle Reynolds number and the critical Reynolds numbers for lift, sliding and rotation, the motion of the particle can be predicted. In Fig. 3, results are shown for the case of glass particles on a glass substrate, subjected to an air flow (p = 1 atm, T = 293 K) with a shear rate of 2 × 106 s−1 at the interface. The intersection points of the different Reynolds lines with the straight particle Reynolds number line indicate the critical particle diameter above which the corresponding motion becomes possible. Particle rotation becomes relevant at 1.6 µm, while particles larger than 80 µm will start sliding (and rotating). Complete lift occurs for particle diameters larger than 100 µm. Rotation is thus the most probable type of motion when dealing with micrometer-sized particles. When a particle starts

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Figure 3. Reynolds numbers in air flow with 2 × 106 s−1 shear rate at the interface for glass particles on glass substrate. The intersection point of the particle Reynolds line with one of the other lines (lift, sliding, rotation) indicates for which particle diameter (and larger) this type of motion (lift, sliding, rotation) is possible. Particles larger than 1.6 µm start rolling, at diameters of 80 µm and above, the sliding type of motion is possible and for diameters above 100 µm, particles are lifted. Particles smaller than 1.6 µm do not move.

moving (rotating), the adhesion interaction decreases since the distance between particle and surface increases. Another effect is that due to rotation the contact area decreases, and thus also the attraction between the particle and surface. Rotation was found to be the dominant mechanism for particle removal in the diameter range investigated. For removal using liquids, it was already known [3, 9– 11] that the rolling forces in water are larger than for sliding and lift. In the current study, this is also shown to be the case for air. For the cases of metal–metal and polystyrene–polystyrene, the graphs are shown in Fig. 4. The critical diameters in the case of metal–metal are close to those calculated for the glass particles. The polystyrene particles are much more strongly attached to the surface, and flow interaction only becomes relevant for larger particle diameters, even when ignoring electrostatic interaction. The material properties used for the different cases, glass, steel and polystyrene (PSL), are summarized in Table 1. 4. Experimental Validation The particle diameters calculated from the expressions derived in the previous section are validated with experimental data. These data are obtained from a flow-cell

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Figure 4. The critical diameter for rotation for different materials. The cases of steel particles on a steel surface and PSL (polystyrene) on PSL are shown, in air flow at 2 × 106 s−1 shear rate. Table 1. Material properties (surface and particle of the same material). Shown are average values from various sources, as used in the calculations Property

Units

Glass

Steel

Polystyrene

Density Hamaker constant Poisson’s ratio Young’s modulus Static friction coeff.

kg/m3 10−20 J – 1010 Pa –

2500 6.5 0.29 8.0 1.0

8000 8.0 0.3 20 0.75

1100 6.6 0.29 0.31 0.50

experiment, in which different wall shear rates can be obtained by changing the flow through the cell. 4.1. Experimental Set-up The set-up used for the experiments consisted of a rectangular channel, as shown in Fig. 5. The width of the channel is much larger than its height, which is only 10 mm. In this geometry, a well-defined flow over the lower surface can be obtained. In the bottom plate, a stainless steel sample holder can be inserted. The metal housing and sample holder are grounded, so electrostatic forces are not relevant during the experiments. The flow is created using a pump which is connected to the top side of the set-up (see Fig. 5). The experiments were performed in a conditioned room with

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Figure 5. The left image shows the design of the flow-cell, on the right the actual set-up. The air flows in through the slit on the left side, and is removed with a pump at the top. The sample is positioned at the bottom of the slit during measurement.

constant relative humidity (45%) and temperature (22◦ C). During the experiment, the sample with particles is subjected to a uniform and constant air flow. Arizona Test Dust was used, which consists of 5–10 µm-sized quartz particles. The particles were distributed homogeneously over the sample holder prior to the experiments. The removal of particles took place within the first second of the flow. The duration of the flow in the experiments varied between 2 and 5 s at atmospheric pressure. Before and after the experiments, the positions and number of particles were measured and counted. 4.2. Shear Rates The turbulent velocity profile (Re > 104 ) in the cell is calculated with Computational Fluid Dynamics (CFD) using the k–ε model. The total mass flow measured in the experiments was used as an input parameter for the simulations. At the wall, a no-slip condition was applied. The calculated velocity profile in the wall region was found to be linear with the distance from the wall (for distances below 0.1 mm in a 10 mm high channel). The shear rates obtained with the set-up range from 7 × 104 up to 2 × 106 s−1 . All particle diameters investigated in the experiments are well within the linear part of the velocity profile. The velocity gradient can thus be taken constant for each individual experiment in the Reynolds number calculations for the particle motion. Since the region of interest is within the viscous sublayer, where the flow can be treated as laminar, the effect of turbulent bursts is not included in the current model. For the different shear rates, the residual percentage of particles on the sample was determined after being subjected to the flow. The particles were counted in 5 diameter classes, 0–5, 5–10, 10–20, 20–50 and 50–100 µm. The residual percentages from the experiments are plotted at the centers of the diameter classes in Fig. 6.

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Figure 6. The residual percentage versus particle diameter is shown for shear rates ranging from 7.89 × 104 s−1 to 1.93 × 106 s−1 . Apart from the curvature at the ends of the lines, a straight line can be fitted though the centre parts.

4.3. Characteristic Particle Diameter From the graph shown in Fig. 6, the d50 value is calculated, which is defined as the diameter for which 50% of the total amount is removed due to the flow (50% residual). For the shear rates corresponding to the experimental conditions, the critical diameters for lift, sliding and rotation are determined with the model. The critical diameters and the experimentally obtained d50 are shown in Table 2. The choice of 50% was based on a symmetrical distribution of the adhesion energy around the most probable van der Waals attraction energy. The diameters calculated for drot are close to the d50 values from the experiments. The difference is only a few micrometers at maximum. All values are underpredicted (drot < d50 ) with a maximum deviation of 33% (except for the last case with highest shear rate). The critical diameters calculated for lift (and sliding) are about three orders of magnitude larger than the diameters of the particles that were removed in the experiments. Apparently, the removal in the experiment was due to rotation, and not due to lift or particle sliding. The diameters drot and d50 were determined for all individual cases of different shear rates. The thickness of the viscous sublayer is determined for each of the 7 different shear rate cases. For the shear rate of 7.89 × 104 s−1 , the thickness of the viscous sublayer equals 70 µm, which is about three times larger than the calculated drot = 25.6 µm particle diameter. For the maximum shear rate of 1.93 × 106 s−1 , the critical particle diameter is well within the 14 µm thick viscous sublayer. The particles that are smaller than the critical diameter will only experience the laminar

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Table 2. Calculated (dlift, rot ) and experimental (d50 ) diameters for different shear rates Shear rate (1/s)

dlift (µm)

drot (µm)

d50 (µm)

7.89 × 104 1.33 × 105 2.20 × 105 2.62 × 105 3.21 × 105 3.53 × 105 1.93 × 106

1740 1047 661 566 473 436 101

25.6 16.3 10.5 9.1 7.6 7.0 1.7

27 19 14 12.6 11.0 10.5 4.1

Figure 7. Both the calculated critical diameter for particle rotation drot and the experimentally obtained diameter d50 (50% of the particles are removed) are shown for different shear rates. The model and the experimental data are in good agreement. The model prediction is, on average, only 3 µm below the experimentally observed diameter, which makes the model a safe estimate for particle movement probability.

flow regime. The assumption of a linear velocity profile is thus valid for the cases investigated. The systematic under-prediction of drot compared to the experimental value for d50 is shown in Table 2. When plotting these diameters versus the corresponding shear rates, both the model and experimental results show the same trend, as shown in Fig. 7. The small under-prediction by the model is clearly visible in the graph. 5. Discussion In a study by Cardot et al. [4], the removal probability is derived based on wall shear stresses instead of wall shear rates. For a 20 micrometer particle, half of the particles are removed at a shear rate of 105 s−1 , as indicated in Fig. 7. This shear

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rate corresponds to a shear stress of about 2 Pa, which is of the same order as the 3.7 Pa shown by Cardot et al. [4] assuming 40% humidity. Note that the 3.7 Pa was obtained with glass particles on a glass substrate, while in the current research quartz particles on stainless steel were investigated. The model proposed here is based on a fixed value for the van der Waals adhesion force. In reality, there is a distribution in adhesion energy between particle and surface, and thus also in attraction force. Due to surface roughness, the particleto-surface distance becomes slightly larger, which implies a lower attraction force. When a small particle fits into the roughness of the surface, the attraction will be higher than expected. For non-spherical particles with a larger contact area, the attraction will be higher than that calculated based on the diameter. The average value of d50 is a good measure for the diameter corresponding to the average interaction force. The removal of particles in the experiments matched the diameter for the rotation mechanism; the diameters calculated for lift and sliding are much larger than the particle diameters that were removed. For spherical particles on a flat surface, the particles start rotating quite easily. For surfaces with roughness, the obstacle or barrier height determines the point of rotation. The surface roughness makes rotation more difficult, and hence larger critical particle diameters will be obtained. For small size particles, the diameter is underestimated by a few micrometers. Since only van der Waals interaction is included for the adhesion, there is a systematic difference between experimental results and the model predictions. The model assumes no surface roughness, while in reality the surfaces of both the substrate and the particles are not perfectly smooth. Small particles tend to settle in the surface imperfections, where the attractive force is higher. Since the experimental set-up used was grounded, and no liquids are involved, both Coulomb and double layer electrostatic effects can be neglected. The capillary effect is not dominant, but might explain the difference between the model and experimental results for smaller particle diameters. The experiments were performed at a relative humidity of 45%. The study by Cardot et al. [4] shows that at this humidity, the removal efficiency decreases. Since the capillary force is not fully developed, the removal at 45% is only slightly lower compared to the 30% relative humidity situation.

6. Conclusions A simple model for particle release from a flat surface is proposed. The critical diameter for removal is calculated based on the forces induced by a gas flow near the surface. The model predictions show good agreement with the experimentally obtained results. The difference, which is within a few micrometers, is most likely due to the capillary sticking of the particles and the variation in adhesion energy over the particles.

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Acknowledgements The authors wish to thank Ruud Schmits and Wim Peterse for performing the experiments, Arnout Klinkenberg for the CFD calculations on the flow distribution and Anton Duisterwinkel for useful discussions. This work is partially financially supported by VSR (Dutch Association for Cleaning Research). References 1. P. Schmitz and J. Cardot, in: Particles on Surfaces 7: Detection, Adhesion and Removal, K. L. Mittal (Ed.), pp. 189–196. VSP, Utrecht (2002). 2. A. A. Busnaina and H. Lin, in: Proc. IEEE International Symposium on Semiconductor Manufacturing Conference, pp. 272–277 (2002). 3. F. Zhang, A. A. Busnaina, M. A. Fury and S. Wang, J. Electron. Mater. 29, 199 (2000). 4. J. Cardot, N. Blond and P. Schmitz, J. Adhesion 75, 351 (2001). 5. G. M. Burdick, N. S. Berman and S. P. Beaudoin, J. Nanoparticle Res. 3, 455 (2001). 6. W. Chen, J. Sedimentary Res. 73, 714 (2003). 7. A. J. Goldman, R. G. Cox and H. Brenner, Chem. Eng. Sci. 22, 653 (1967). 8. Y. Mamonoi, K. Yokogawa and M. Izawa, J. Vac. Sci. Technol. B 22, 268 (2004). 9. A. A. Busnaina, H. Lin and N. Moumen, in: Proc. IEEE International Symposium on Semiconductor Manufacturing Conference, pp. 328–333 (2000). 10. F. Zhang and A. A. Busnaina, Appl. Phys. A 69, 437 (1999). 11. F. Zhang, A. A. Busnaina and G. Ahmadi, J. Electrochem. Soc. 146, 2665 (1999).