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Numerical Study on the Adhesion and Reentrainment of Nondeformable Particles on Surfaces: The Role of Surface Roughness and Electrostatic Forces Christophe Henry,*,†,‡ Jean-Pierre Minier,*,† and Gregory Lefevre‡ † ‡
Fluid Dynamics, Power Generation and Environment, EDF R&D, 6 quai Watier, Chatou 78401, France LECIME, CNRS-Chimie ParisTech, UMR 7575, 11 rue Pierre et Marie Curie, Paris 75005, France
bS Supporting Information ABSTRACT: In this paper, the reentrainment of nanosized and microsized particles from rough walls under various electrostatic conditions and various hydrodynamic conditions (either in air or aqueous media) is numerically investigated. This issue arises in the general context of particulate fouling in industrial applications, which involves (among other phenomena) particle deposition and particle reentrainment. The deposition phenomenon has been studied previously and, in the present work, we focus our attention on resuspension. Once particles are deposited on a surface, the balance between hydrodynamic forces (which tend to move particles away from the surface) and adhesion forces (which maintain particles on the surface) can lead to particle removal. Adhesion forces are generally described using van der Waals attractive forces, but the limit of these models is that any dependence of adhesion forces on electrostatic forces (due to variations in pH or ionic strength) cannot be reproduced numerically. For this purpose, we develop a model of adhesion forces that is based on the DLVO (Derjaguin and Landau, Verwey and Overbeek) theory and which includes also the effect of surface roughness through the use of hemispherical asperities on the surface. We first highlight the effect of the curvature radius on adhesion forces. Then some numerical predictions of adhesion forces or adhesion energies are compared to experimental data. Finally, the overall effects of surface roughness and electrostatic forces are demonstrated with some applications of the complete reentrainment model in some simple test cases.
’ INTRODUCTION Tremendous challenges are associated with fouling of heat exchangers in the automotive industry, as well as in food processing or petrochemical industries, since fouling generally decreases the efficiency of the apparatus. In a pressurized water reactor (PWR),1 fouling of heat exchangers in the secondary circuit can occur due to the presence of metallic oxide particles (which stem from the corrosion of pipes), and this fouling reduces heat transfer efficiency. In order to address the challenges associated with fouling, there is an ongoing need to understand the fouling process, to design a physical model, and to perform numerical simulations of the fouling process. Particulate fouling can be divided into various underlying mechanisms, such as particle deposition and particle reentrainment. Particle deposition is the first mechanism in the fouling process, and it corresponds to the transport of particles by the fluid toward walls where they can interact with the surface. Particle reentrainment occurs once particles adhere to a surface: hydrodynamic forces exerted on adhering particles can overcome adhesion forces, leading to their removal from the surface. From this simple overview of the early stages of fouling, it can be noticed that particulate fouling is related to the coupling of two phenomena: hydrodynamic transport of particles (fluidparticle r 2011 American Chemical Society
interactions) and physicochemical relations (intersurface interactions). Since hydrodynamic transport and physicochemical interactions occur at two widely different length scales, we consider the fouling process to be accurately described by a two-step process: the transport step and the attachment step. In previous studies, the deposition mechanism has been modeled accordingly: the transport step was modeled using a stochastic Lagrangian model for hydrodynamic transport,2,3 while the attachment step was described using the DLVO theory (which takes into account both van der Waals and electrostatic interactions4) and accounting for surface roughness through the use of hemispherical asperities.5 Numerical predictions were shown to be quite accurate, and the deposition mechanism is thus considered to be well-reproduced. Aim of the Present Study. In this paper, we focus our attention on particle resuspension, the second mechanism of fouling. In a previous paper from Guingo and Minier,6 a new stochastic Lagrangian model to simulate particle resuspension has been proposed, mainly aiming at addressing the role of Received: September 19, 2011 Revised: November 21, 2011 Published: November 22, 2011 438
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surface roughness on resuspension. Surface roughness was shown to play a role in the scenario of particle resuspension, but also in the distribution of adhesion forces. However, the model proposed was based on a simplified 1D formulation of surface roughness, and adhesion forces were calculated using both JKR (JohnsonKendallRoberts7) theory and a Hamaker approach (depending on the presence of asperities in contact with the particle). Since adhesion forces were determined using only van der Waals forces, this model accounts neither for the pH-dependence of adhesion forces, which has been observed experimentally,810 nor for the pH-dependence of particle resuspension.11,12 In addition, the distribution of adhesion forces were directly obtained by assuming a log-normal law with mean and standard deviation equal to the evaluation of surface forces obtained using a Hamaker approach. With respect to this first modeling attempt, the aim of the present paper is 3-fold: (1) to have a more detailed description of surface roughness, which would allow the distribution of adhesion forces to be generated by variations in surface roughness properties; (2) to describe adhesion forces using a single model, which would have to be in line with the model developed previously for the deposition mechanism;5 (3) to include electrostatic forces in the calculation of adhesion forces, so as to be able to reproduce the pHdependence observed experimentally on adhesion forces and resuspension. The present paper is divided in two parts. The first part is devoted to the description of adhesion forces between rough surfaces, while the second part deals with the reentrainment of particles from rough surfaces by hydrodynamic forces. In each part, we first describe the numerical model which is used and then present comparisons between experimental data and numerical predictions.
dealing with rough surfaces, multiasperitycontact adhesion models are used.17 In such models, the adhesion force is obtained by summing the contact force for each asperity in contact with the particle. Another method to calculate the adhesion force between two surfaces is to use the Hamaker approach (or van der Waals approach): the adhesion force is given by the integration of molecular interaction over the whole volume of the two interacting bodies. The difficulty here is to choose a correct description of surface roughness, as pointed out in Eichenlaub et al.:18 some models are based on a finite element method (FEM),19 while other models consider fractal surfaces. Fourier transform are also used20 as well as hemispherical asperities.2126 The choice of the appropriate description often boils down to a balance between computational costs and model precision: it can be easily understood that a model using an FEM description of surface roughness is more precise than a simple model using hemispherical asperities. It is also less restrictive (since intricate geometries can be reproduced numerically) but is much more computationally demanding than a model based on hemispherical asperities. The advantage of the Hamaker approach is that it takes into account noncontact interactions, and it also allows the calculation of the force at various separation distances. The drawback of this method is that surface deformations are generally not taken into account. However, both methods describing the adhesion force generally account solely for van der Waals interactions between surfaces. Therefore, any dependence of adhesion forces on electrostatic forces cannot be reproduced numerically. Only a few studies8,10,27 do include the two aspects of the DLVO (Derjaguin, Landau, Verwey, and Overbeek) theory4 in the calculation of the adhesion energy, a theory which accounts for both van der Waals forces and electrostatic forces. Some studies also introduce specific non-DLVO forces, such as structural forces,9 acidbase forces28 or specific surface reaction,29 which may occur depending on the experimental conditions. Present Model. Following a previous work made on deposition,5 we have chosen to use a Hamaker approach. Surface deformations are indeed not likely to play a predominant role in the resuspension mechanism of particles in the industrial cases of interest (the elastic modulus of surfaces being rather high). Therefore, we have chosen to use a model that is only valid for nondeformable surfaces (Tabor’s parameter μT e 1). Since the aim of the study is to be able to predict the whole fouling process in industrial applications, the reentrainment of particles is only one phenomenon among others. Therefore, even within the Hamaker approach, the use of complex approaches such as FEM (which are computationally expensive) to calculate adhesion forces seems ill-adapted in our case and we choose to model adhesion forces through a simplified approach based on hemispherical asperities. This allows fast evaluation of the adhesion force distribution through Monte Carlo simulations, and such a model is coherent with the model previously designed for the deposition mechanism.5 Following the previous work on reentrainment made by Guingo and Minier,3 surface roughness is also described with two-scale roughness features: asperities in the nanometer range and asperities in the micrometer range are considered. The scheme used to determine adhesion forces or adhesion energies is 2-fold: the number of particleasperity contacts is first evaluated using a Poisson distribution, with a mean equal to the number of asperities within an interacting area (representative
’ APPROACH TO SIMULATE ADHESION FORCES In this first section, a new model to determine the adhesion force or the adhesion energy between a particle and a rough surface is proposed and some numerical predictions are compared to experimental data. Numerical Model. Existing Models. A review of adhesion models has been made recently by Prokopovich et al.13 where it was been pointed out that two different methods are available to determine the adhesion between ideal smooth surfaces. The first method is based on surface energies (also called contact mechanics), as is the case in the JKR (JohnsonKendallRoberts) theory,7 the DMT (DerjaguinMullerToporov) theory,14 or the MP (MaugisPollock) theory.15 Whereas the JKR theory accounts for surface deformations (with respect to the Hertz model) and assumes that adhesion forces act only within the contact area between surfaces, the DMT theory includes noncontact forces. The limitations of these theories are well-known, and the range of applicability of the JKR and DMT models has been studied by Tabor:16 he introduced a dimensionless coefficient μT3 = RpartΔγ2/(E2z03) {Rpart being the particle radius, Δγ the surface energy, z0 the equilibrium separation of surfaces, E = 4/3[(1 ν12)/E1 + (1 ν22)/E2]1 the composite Young modulus, and ν the poisson ratio}. This Tabor’s parameter is a measure of the ratio between surfaces adhesiveness and surfaces stiffness. The JKR model should be applied when μT . 1 (noncontact forces can be neglected), whereas the DMT model is valid for μT e 1. When 439
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Figure 1. Contact zone, corresponding to the zone where asperities will be in contact with the sphere.
of asperities playing a role in the calculation); then, assuming interaction energies to be additive, the adhesion energy can be calculated by summing each contribution to the whole interaction. The same Hamaker approach is thus followed to evaluate the adhesion force in different configurations, whereas previously,6 the JKR theory was used in the sphereplate case and a Hamaker approach was used in the case of sphereasperity contacts. Number of Contacts. The first step toward a calculation of contact forces is to evaluate the number of particleasperity contacts. As emphasized by Katainen et al.,30 the relative size of the adhering particle and surface features is one of the key parameter in the determination of adhesion forces, since it rules multiasperity contacts. In their model, the number of asperities in contact with the particle was given by AintFasp, Aint being the interacting area and Fasp the asperity density. In our case, a rough surface is modeled using hemispherical asperities (radius Rasp) placed on a smooth plate. Thus, an asperity can be in contact with the approaching sphere (radius Rpart) if it is inside the contact region (surface Scont) given by geometric considerations (see Figure 1a): Scont ¼ πRcont 2 ¼ πRasp ðRasp þ 2Rpart Þ
indeed to less than 10% of the whole interaction force. Therefore, for multiple contacts, the average number of asperityparticle contacts is given by ef f ef f 2 ¼ 1 þ Fasp πðRcont Þ Nav
¼ 1 þ Fasp π2z0 ð2Rasp þ 2Rpart þ 4z0 Þ
ð2Þ
Equation 2 is the sum of two terms: the first term (unity) stands for the presence of at least one asperity under the particle (since Ncont > 1 from the first estimation), and the second term yields the number of asperities that are close enough to the particle. Once again, the number of asperityplate contacts Neff cont is obtained using a Poisson distribution with mean Neff av . As pointed out earlier, we consider surface features described by a two-scale roughness: asperities in the micrometer range (which will be later referred to as large-scale asperities) and asperities in the nanometer range (later referred to as small-scale asperities). The number of large-scale asperities in contact with the surface is first determined using the method described above. Then, if no large-scale asperities contact the particle, the same procedure is used to determine the number of contact between the particle and small-scale asperities. Otherwise, it is assumed that particleplate contact is no longer possible, and the number of small-scale asperities is obtained considering that these nanoscale asperities cover the large-scale asperities (see Figure 2 describing the numerical scheme). The formulas giving the average number of small-scale asperities lying on large-scale asperities and in contact with the particle are obtained by geometric considerations similar to those used to give eqs 1 and 2 (details are available in the Supporting Information), and then, the number of small-scale asperities in contact with the particle is determined using a Poisson distribution. Adhesion Energy and Adhesion Force. Once the number of particleasperity contacts is determined, the adhesion energy is obtained using the DLVO theory and assuming interaction energies to be additive:
ð1Þ
The average number of asperities (Nav) within this contact surface is obtained by multiplying the contact surface by the density of asperities, Fasp (in m2). The morphology of rough surfaces being rather chaotic, the number of asperities in contact with the particle (Ncont) is obtained using a Poisson distribution with mean Nav. It can be easily understood that the average number of asperities with a size comparable to the particle diameter will be quite small (around one or two or even less), but the average number of small-scale asperities on such a contact surface can be quite large (depending on the density of smallscale asperities). Since our model does not account for surface deformation, the number of contacts between a sphere and a rough substrate cannot reach values as high as a few hundred. Therefore, in the case of multiple contacts (Ncont > 1), the number of asperities that really play a role in the adhesion force has to be extracted from this first rough estimation. To do so, we first assume that one asperity lies right under the particle and we then consider only the asperities within a certain distance (see Figure 1b). This approach is similar to the one proposed by Rabinovich et al.,21 who ignored asperities lying too far from the particle: since van der Waals forces (which are the predominant forces in adhesion) decrease rapidly with the distance, asperities located at a distance greater than 3z0 (z0 being the cutoff distance used to calculate the interaction energy or the adhesion force) are assumed to be negligible. Beyond 3z0, the force contributes
adh adh adh ¼ ð1 Scov ÞUSP þ Ncont USA USR
ð3Þ
which reduces to the following equation since VDW forces (being volume forces) are predominant for contact interactions adh adh adh ¼ USP þ Ncont USA USR
ð4Þ
where Uadh SP is the interaction between the sphere and the plate (which may be separated by an asperity) and Uadh SA is the interaction between a sphere and an asperity. The interaction 440
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Figure 3. Topographical parameter of a rough surface.
Figure 4. Curvature radius.
(here 45 000 times) so as to be able to extract the whole distribution of adhesion forces (and not only one value corresponding to a single particular case of particleplate contact). This provides information on the mean value, minimum value, maximum value, and standard deviation of the adhesion force and adhesion energy. Moreover, it can be seen that distribution of adhesion forces will be obtained by variations in both the asperity radius and the number of contacts (which is linked to the coverage—or density—of asperities on the surface). Curvature Radius. From the previous description of the model used in this study, it appears that two up to four parameters are required to describe surface roughness: the asperity height and the surface coverage for the two possible asperity scales. In many experimental data, the characterization of surface roughness is made using an AFM, from which the rms roughness (which corresponds to the quadratic mean height of surface roughness above a certain mean plate) and the peak-to-peak distance (λpp) (mean distance between two successive asperities) are extracted (Figure 3). In some models, such as the one from Rabinovich et al.,21 the rms roughness is considered to be representative of surface roughness. However, in some cases, surface roughness cannot be described only with the rms roughness, since surface protrusions appear to be ellipsoidal: for example, in the paper from Prokopovich et al.,17 the asperity shape was described using a parabolic equation. Therefore, in the present study, the asperity height required to perform calculations is assumed to be equal to the curvature radius, and the surface coverage is related to the peak-to-peak distance by Scov = πRasp2/λpp. The curvature radius of an ellipsoidal asperity is given by geometric considerations (see Figure 4). In the case where the asperity height is small compared to its length characterizing the particle size on the surface (as depicted in Figure 4a), the use of a hemispherical asperity with a radius equal to the asperity height will lead to an underestimation of the adhesion force. van der Waals forces are indeed the predominant forces in the adhesion phenomena, and since the range of van der Waals forces is quite small, only the two facing volume will play a role in the value of the adhesion force. It can easily be seen in Figure 4a that the facing volume of a spherical asperity is completely different than the facing volume of an ellipsoidal asperity. Therefore, instead of using the asperity height, the adhesion force is estimated using the curvature radius,
Figure 2. Outline of the numerical scheme used for adhesion forces between rough surfaces.
energy between two contacting surfaces is evaluated at a separation distance z0, which is here considered to be equal to z0 = 0.165 nm independently of surface properties (corresponding to the cutoff distance used in the evaluation of adhesion forces4). Formulas used for van der Waals interactions and electrostatic interactions have been described in a previous paper.5 van der Waals interactions are calculated taking into account retardation effects (formulas are thus also valid at larger separation distances): VDW interactions are proportional to the Hamaker constant AHam, to the separation distance, and to the particle radius. Electrostatic interactions are obtained using the formula from Bell et al.,31 valid at all separations, with the amelioration for large ζ-potential from Ohshima et al.:32 EDL interactions require the knowledge of the double layer electric potential, which is here assumed to be equal to the ζ-potential (more details on formulas for VDW and EDL interactions are provided in the Supporting Information). The adhesion force is then estimated using the Newton’s difference quotient with ε = 1011 m: adh ¼ FSR
adh adh USR ðz0 þ εÞ USR ðz0 Þ ε
ð5Þ
The procedure described above to calculate the number of asperityparticle contact is repeated a large number of times 441
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Figure 5. Adhesion between 10 μm polystyrene particles and SiO2 coatings: (a) blue circle, experimental data;19 green dashed line, predictions with Scov = 75%; and red line, predictions with Scov = 60% (Rasp = 17 nm); (b) blue circle, experimental data;19 green dashed line, predictions with Rasp = 125 nm; and red line, predictions with Rasp = 170 nm (Scov = 75%).
which can be related to the asperity height (h) and characteristic length on the surface (D) by Rasp ¼
h2 þ D2 2h
spherical asperities with a constant diameter (34 or 250 nm) and with a surface coverage equal to 78.5% (this is the maximum value of the surface coverage corresponding to a surface entirely covered by spherical asperities placed in a cubic packing). In this experiment, the Tabor’s parameter is equal to 4.4: this value is slightly higher than 1 (showing that small deformations may occur), but it should be borne in mind that, since particleasperity interactions occur, the adhesion force will be smaller than in the case of particleplate interactions (leading to a Tabor’s parameter close to 1.6 for 250 nm asperities and 0.8 for 34 nm asperities). Therefore, even if some deformations can take place, their value will remain small enough to allow the use of a Hamaker approach. It can be seen in Figure 5a that the numerical predictions exhibit the same trend as the experimental observations for wafers with 34 nm coatings, but the minimum value of the adhesion force is overestimated by the model. The staircase shape of the cumulative distribution function is due to the discrete values of asperityparticle contacts, each number corresponding to a specific adhesion force. For a 78.5% surface coverage, the average number of contacts Nav is equal to 9.13. Slightly lowering the surface coverage would decrease Nav, leading to better numerical predictions (see the red line in Figure 5a): numerical predictions using a surface coverage of 60% (which still corresponds to a relatively dense monolayer) are shown to reproduce correctly the observations (a standard deviation of 2 nm in the asperity radius has been introduced in order to smooth the curve). Numerical predictions for wafers with 250 nm coatings underestimate the adhesion force (as depicted in Figure 5b), even if the minimum and maximum values are in agreement with experimental data. Increasing the surface coverage to 90% (related to hexagonal close-packed spheres) cannot explain such a difference. This discrepancy is rather likely to stem from the assumption upon which, when multiple contacts occur, an asperity is first placed right under the particle center in order to evaluate the real number of asperities playing a role in the adhesion. In reality, this first asperity may be shifted to the particle side, leading to a higher probability to have a second contact. To assess the sensitivity of the model to parameters such as the particle radius, a numerical calculation has been performed using asperities of 340 nm with a standard deviation of 40 nm, showing better agreement with the experimental observation.
ð6Þ
Similarly, in the case where the asperity height is larger than its diameter stretch (as sketched in Figure 4b), the curvature radius is assumed to be equal to the diameter stretch, since the volume of the asperity facing the particle seems to be close to the one obtained using the diameter stretch. Comparison to Experimental Data. In this part, numerical predictions obtained using the previously described model are compared to several experimental data. Among all the parameters required to perform simulations, only the influence of parameters related to surface roughness (asperity size and coverage) has been studied, whereas parameters related to surface properties (Hamaker constant, potential) and to solution conditions (ionic strength) are considered as constant and given by experimental data. In order to do so, realistic evaluations of the roughness parameters have been extracted from available experimental data and have been varied accordingly. The influence of surface roughness on the distribution of adhesion forces is first shown to be well captured by the model. Moreover, the effect of the curvature radius on the adhesion force is highlighted in some relevant cases. Then, the model is shown to be able to reproduce the influence of electrostatic forces on adhesion forces. Influence of Surface Roughness. There is now a wealth of experimental data dealing with the influence of surface roughness on adhesion forces. From these experimental studies, it is wellknown that surface roughness induces a wide distribution of adhesion forces, since surface features of various sizes induce different adhesion forces and since the chaotic morphology of surface roughness results in a random nature of surface contacts. (a) In their experiment, Zhou et al.19 used an AFM to measure the adhesion force between 10 μm polystyrene particles and silicon wafers coated with 34 and 250 nm SiO2 nanoparticles. Numerical calculations have been performed using a Hamaker constant of 6.5 1020 J. Since the experiment was made in air, electrical double layers would not exist and it is thus assumed that the DLVO theory is described by van der Waals forces only. Knowing that nanoparticles have built a densely packed monolayer on the wafer surface, we consider a smooth plate covered by 442
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Figure 6. Adhesion between 10 μm polystyrene particles and aluminum substrates: (a) blue circle, experimental data;25 red line, predictions with Rasp = 250 nm (Scov = 0.5%); (b) blue circle, experimental data;25 red line, predictions with Rasp = 850 nm; green dashed line, predictions with Rasp = 500 nm; yellow dasheddotted line, predictions with Rasp = 1.2 μm (Scov = 75%).
used. As in the first case, the use of the rms roughness does not provide accurate results, since asperities appear ellipsoidal on the roughness profile provided in their paper.25 An evaluation of the curvature radius from the roughness profiles is intricate, since surface features of various sizes cover the whole surface. Numerical predictions have been made with a mean roughness of 850 nm and a standard deviation of 400 nm (see Figure 6b), showing good agreements with experimental data. Increasing the mean asperity radius leads to a higher mean adhesion force (as depicted in Figure 6b, where results from predictions using a mean roughness of 500 or 1200 nm are drawn), whereas a larger standard deviation of the curvature radius enlarges the spread in adhesion forces. Therefore, in order to correctly estimate the adhesion force between rough surfaces, detailed statistical information on the curvature radius of surface features has to be available. (c) Another experiment has been carried out by Prokopovich et al.17 on adhesion between a rough PBT particle (diameter around 60 μm) and PVC or stainless steel rough substrates. The PVC substrate, being highly deformable (its elastic modulus is only equal to 33.5 MPa), will not be studied here, since the Tabor’s parameter reaches a value as high as 209 (showing that significant deformations of surfaces occur). Attention will thus be focused on the adhesion of PBT particles on stainless steel (where the Tabor’s parameter is equal to 1.4 between rough surfaces). From the surface energies measured by the authors and using the formula from Israelachvili4 linking surface energies γ and Hamaker constants (AHam = γ24πz02), an estimation of the Hamaker constant for the system considered can be obtained: AHam = 10.3 1020 J. No electrostatic interaction occurs in the present case. The authors have measured the asperity height of the substrate as well as its curvature radius, using a parabolic equation: the distribution of curvature radius was found to follow a Weibull distribution with a scale parameter equal to 0.162 μm and a shape parameter of 1.643.17 The AFM images of the stainless steel substrate showing a highly rough surface (the measured asperity density 5.32 1012 particles/m2 is also high), the surface coverage is considered to be equal to 50%. However, the roughness on PBT particles is not described. Nonetheless, a profile image of a PBT particle gives some insight into the roughness characteristics:17 ellipsoidal asperities with a curvature radius of 36 nm cover around 50% of the surface. Therefore, in
(b) In a similar experiment, Zhou et al.25 measured the adhesion force between 10 μm polystyrene particles and aluminum substrates with various roughness characteristics using both AFM and centrifugal detachment techniques. The Hamaker constant between polystyrene and aluminum substrates across air is equal to 17.1 1020 J. In this case, the Tabor’s parameter between rough surfaces is around 4.8, showing that some deformation may exist. Nonetheless, satisfactory predictions can still be obtained using the Hamaker approach. First, the rms roughness of aluminum polished substrate has been measured and is equal to 6.4 nm, with a peak-to-peak distance of 257 nm, which corresponds to a surface coverage of 0.2%. The experimental observations show a two-peak distribution of the adhesion force (see Figure 6a): the first peak, around 15 nN and with a frequency of 18%, is related to the particle asperity contact, whereas the second peak, around 155 nN, is due to the sphereplate contact. This cannot be reproduced numerically using these parameters, since the adhesion force between one particle and one asperity of 6.4 nm is equal to 2.5 nN (and more than one contact between the particle and asperities is rather unlikely due to the low surface coverage). Numerical predictions of the first peak thus underestimate the adhesion force measured experimentally. Yet, the second peak, corresponding to the sphereplate contact, is well-reproduced by the numerical predictions. Therefore, the origin of the discrepancy between predictions and experiment on the value of the first peak in the adhesion force is probably due to a misestimation of the surface roughness. On the roughness profiles provided in the paper from Zhou et al.,25 asperities on the surface appear to be ellipsoidal, and thus, the corresponding curvature radius has to be evaluated. However, reading the curvature radius on such roughness profiles is cumbersome. For instance, it was found that the curvature radius is roughly equal to 250 nm, with a standard deviation around 32 nm, and the corresponding surface coverage was close to 0.5%. Using such parameters yields numerical results that reproduce the general trends observed experimentally (see Figure 6a). It can be noted that the first peak is a bit lower than in the experiment, showing that the evaluation of the curvature radius has still to be improved. Second, another aluminum surface with an rms roughness equal to 17.7 nm and a peak-to-peak distance of 157 nm has been 443
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It can be seen in Figure 8a that the adhesion between a rough sapphire particle and a sapphire plate can be correctly reproduced by the numerical model, particularly the small values of adhesion forces. However, the spread in predicted adhesion forces is larger than in the experiment. This may be due to the mean asperity radius used in the calculation. For instance, using a mean radius of 25 nm (which corresponds then to a surface coverage of 24%) and a standard deviation of 10 nm yields predictions closer to experimental observations (see the dotted line in Figure 8a). It should be noted that the presence of largescale asperities on the sapphire substrate also plays a role: since at least one large scale asperity (1 μm) faces the surface, the number of small scale asperities is lower than what can be expected using only a smooth plate and 29 nm asperities (which would yield higher values of the adhesion forces). The case of a sapphire particle in contact with a polished ceramic surface has been modeled using large-scale asperities of 170 nm, with a standard deviation of 50 nm covering 6.4% of the surface (corresponding to the large-scale asperities measured on the polished ceramic substrate), and small-scale asperities of 30 nm, with a standard deviation of 20 nm covering 41% of the surface (representing the small-scale asperities of the ceramic substrate and the asperities on the particle). It can be seen in Figure 8b that satisfactory results are obtained. One can argue that the minimum value of the adhesion forces still differ from the experimental data, but this is probably due to the fact that we have chosen (for the sake of simplicity) to model surface roughness only on the plate. In reality, 32 nm asperities on the plate interact with 29 nm asperities on the particle, leading to lower minimum values of the adhesion force (around 3.5 nN). Finally, we have modeled the ceramic substrate with a largescale roughness of 213 nm and a standard deviation in asperity size of 100 nm. The results obtained (see dotted line in Figure 8c) overestimate the measured adhesion forces. This may stem from the poor characterization of the ceramic roughness, since it is only described in the paper with a rms roughness of 213 nm, whereas topography measurements or the surface profile available in the paper shows a large variety of roughness sizes (asperities in the micrometer range are visible, and it seems that asperities in the nanometer range also exist). Surface features as small as 10 nm on the substrate will lower the predicted adhesion force, since they will interact with 29 nm asperities on the particle. A numerical calculation with asperities of 15 nm (standard deviation equal to 10 nm) has been performed (see the red line in Figure 8c), showing better agreement with experimental data and thus highlighting that asperities of roughly 10 nm on the ceramic surface can explain this discrepancy. Influence of Electrostatics. Studies on the influence of electrostatic forces (pH dependence or ionic strength dependence) are scarce.810,27 Among the few experimental data available, some of them deal with the influence of non-DLVO forces, such as surface complexation29 (which influences the surface potential, allowing variations of the potential with deposition time), acid base interactions28 (corresponding to the interaction between surface functional groups, which are either electron donors or electron acceptors, in relation with the hydrophobic/hydrophilic nature of surfaces that results in an additional attractive or repulsive force), or steric forces9 (arising from the geometry of solvent molecule and from their packing between constraining boundaries, which leads to an oscillatory force as described by Israelachvili4). In their paper, Eichenlaub et al.8 studied the adhesion of alumina particles on silicon dioxide substrate in an aqueous
Figure 7. Adhesion between 60 μm PBT particle and stainless steel: blue circle, experimental data;17 red line, numerical predictions.
our numerical model, we consider a two-scale roughness on the smooth substrate: the large-scale roughness (162 nm) corresponding to the substrate roughness and the small-scale roughness (36 nm) corresponding to the particle features. It can be seen in Figure 7 that the numerical predictions obtained using these parameters are quite accurate. The predicted minimum value of the adhesion force is in agreement with the experimental data, and this is only due to the interaction with small-scale asperities (the use of large-scale asperities only will not provide such a low value in the adhesion forces). The predicted maximum value of adhesion is higher than in the experiment, and this can be linked with some misevaluation of the particle surface features (for instance, a surface coverage of 30% only for small-scale asperities will lower the maximum value of adhesion). (d) Audry et al.33 have studied the adhesion between sapphire particles and highly rough alumina surfaces using AFM. The value of the Hamaker constant between alumina surfaces and sapphire particles has been given by the authors and is equal to 4.2 1020 J. The experiments were done in aqueous environment, and the ζ-potential of both surfaces is assumed to be constant (i.e., independent of the separation distance) and equal to 25 mV with a Debye length k1 = 50 nm (thus, some repulsion can occur between surfaces). The Tabor’s parameter between such surfaces is lower than 0.2 (the value of μT in the sphereplate case), highlighting the absence of notable surface deformations. The sapphire particles (46 μm in diameter) were shown to be rough, with a rms roughness of 29 nm and a peak-topeak distance of 90 nm. However, a quick look at one typical section of the particle reveals that some asperities are slender ellipsoidal asperities, leading to a curvature radius of around 1015 nm. Therefore, surface features on sapphire particles are described in the model by a mean roughness of 29 nm with a standard deviation of 15 nm and a surface coverage of 33%. These particles are brought into contact with three different alumina substrates: a sapphire plane (with a rms roughness of 1 nm, but with a curvature radius around 1 μm or larger), a rough ceramic surface (rms roughness of 213 nm and peak-to-peak distance of 3.7 μm, corresponding to a surface coverage of 1%), and a rough polished ceramic surface (rms roughness of 32 and 170 nm with corresponding peak-to-peak distances of 201 and 1190 nm, or surface coverage of 8% and 6.4%). Since both particle and substrate surfaces are rough, they can be modeled using a smooth particle adhering to a rough plate, whose roughness represents the measured roughness of both the plate and the particle. 444
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Figure 8. Adhesion between 46 μm sapphire particles and alumina substrates: (a) blue squares, experimental data;33 red line, predictions with Rasp = 25 nm, Scov = 24%; green dashed line, predictions with Rasp = 29 nm, Scov = 33%; (b) blue squares, experimental data;33 red line, predictions with Rlarge asp = 33 small small large 170 nm, Slarge = 213 nm, Slarge cov = 6.4%, and Rasp = 30 nm, Scov = 41%; (c) blue squares, experimental data; red line, predictions with Rasp cov = 1%, and small large large small small Rsmall asp = 15 nm, Scov = 33%; green dashed line, predictions with Rasp = 213 nm, Scov = 1%, and Rasp = 29 nm, Scov = 33%.
solution at various pH values with an ionic strength equal to 0.01 M (the Tabor’s parameter associated with this system is equal to 0.05 for the sphereplate case, showing that the use of a Hamaker approach is a reasonable assumption). Alumina particles are not spherical,34 but they can be modeled using a sphere with a nominal diameter around 5 μm. The rms roughness on alumina particles has been measured experimentally and equals 8.6 nm, and the surface coverage is assumed to be equal to 78.5% (since the AFM image of an alumina particle shows a highly rough surface). The Hamaker constant of the SiO2waterAl2O3 system is equal to 1.3 1020 J.34 ζ-Potentials of alumina particles and silica substrate were measured experimentally as a function of pH at a constant ionic strength of 0.01 M and results are depicted in Figure 9. A rapid comparison of ζ-potentials provide some information on the expected trend of adhesion forces using the DLVO theory: electrostatic repulsion between surfaces occurs at pH smaller than 2, followed by an attractive force from pH 4 up to 8, and repulsion is expected at pH larger than 10. The results presented in Figure 10a show that both experimental data and numerical predictions are in agreement with the expected trend of adhesion forces. Predictions are also quite accurate for the minimum and maximum as well as the mean value of the adhesion forces, but they are slightly less accurate for the standard deviation of adhesion forces. Moreover, numerical predictions reproduce the influence of the ionic strength on the adhesion force (see Figure 10b), which was studied by the authors.
Figure 9. ζ-Potential of (blue circles) alumina particles and (red x’s) silica substrates with varying pH conditions at a ionic strength of 0.01 M (experimental data from ref 8).
Summary. From this study, we have seen that the distribution of adhesion forces (measured experimentally) can be explained by the coupling of two phenomena due to surface roughness: first, the number of asperityparticle contacts leads to different adhesion forces (multiples of the adhesion force for a single asperityparticle contact); second, the distribution of asperity size is also responsible for variations in adhesion forces, since they are proportional to the Derjaguin prefactor RpartRasp/(Rpart 445
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Figure 10. Comparison of experimental data8 (symbols) and numerical predictions (lines) of the adhesion between 5 μm alumina particle and silicon dioxide surface: black x’s and black solid line, mean value; green triangles and green dashed lines, maximum value, blue circles and blue dasheddotted line, minimum value; red diamonds and red dotted line, standard deviation.
+ Rasp). In addition, taking into account both van der Waals and electrostatic forces allows us to reproduce the evolution of adhesion forces with electrostatic conditions. However, some of the previous comparisons between numerical predictions and experimental data have emphasized the need to consider the curvature radius, and not the rms roughness, when dealing with nonspherical asperities on the surface. It should also be borne in mind that a detailed statistical description of surface features is required to perform numerical predictions that agree quantitatively with experimental results.
process are reminded here. A three-stage scenario for particle reentrainment has been proposed, based on the idea that hydrodynamic drag forces can force particle to roll on a rough surface (see Figure 11): first, if the moment of hydrodynamic forces is high enough to overcome the moment of adhesion forces, the particle starts to roll (or slide) on the surface; second, the particle moves on the surface (eventually on the small-scale surface features); third, upon rocking on an asperity, the particle may be dragged away from the surface if its kinetic energy (considered to be converted from the streamwise to the wallnormal direction) is greater than the adhesion energy. From this overview of the model, it can be deduced that the balance between the moment of adhesion forces and the moment of hydrodynamic drag forces governs the first and second stages of the scenario, while the adhesion energy and the particle kinetic energy control the third stage. The model for adhesion forces described in the first part of this paper provides information on the adhesion force as well as on the adhesion energy. The moment of adhesion forces around a point O is given by:
’ APPROACH TO SIMULATE PARTICLE RESUSPENSION In this section, the new model for adhesion forces, which has been described in the previous part of this paper, is coupled with the resuspension model implemented in Code_Saturne, a computational fluid dynamic (CFD) code.6 Numerical model. Existing models. According to the reviews on particle resuspension,3537 two different classes of resuspension models can be distinguished. • Quasistatic models: In such models, particle motions are governed by the balance between aerodynamic forces or moments and the corresponding contact forces or moments. It has been shown by Ziskind et al.35 that the lift force is unable to account for the resuspended fractions observed experimentally, but that rolling of particles on the surface is the primary phenomena in particle resuspension. A mechanism has thus been suggested, in which particles first roll on the surface, when the drag force moments overcome the contact force moments, and are later resuspended. • Dynamic models: In these models, particle resuspension is governed by an energy accumulation approach. For instance, in the initial RRH (ReeksReedHall) model,38 a particle is resuspended once its cumulated vibrational energy (due to lift forces) is high enough to overcome the adhesive potential well. Present Model. The complete reentrainment scenario used in the present study is a quasistatic model which has been described in a previous article6 and the main principles of the resuspension
MO ðFadh Þ ¼ Fadh a0
ð7Þ
where a0 is the distance between the vertical line going through the particle center and the point O. To obtain the moment of adhesion, the point O is taken as the furthest contact point between the particle and an asperity in the downstream direction. In our case, since we use a stochastic approach to determine the adhesion force, the exact position of each asperity is not known. Therefore, to evaluate the adhesion moment, the distance a0 is randomly chosen inside the contact area (given by eq 1). In the event of particleplate contact, the distance around which the particle may rotate is assumed to be equal to a deformation radius extracted from the Hertz theory.39 This deformation radius is given by a0 ¼ ðFadh Rpart =EÞ1=3 where E is the composite Young’s modulus:40 !1 4 1 ν1 2 1 ν2 2 E¼ þ 3 E1 E2 446
ð8Þ
ð9Þ
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Figure 11. Resuspension model: a three-stage scenario.
The deformation radius obtained is generally small, and it is close JKR to the deformation radius √ extracted from the JKR theory, a0 = 1/3 (12πγRpart/E) = a0 3 (γ being the surface energy). As far as hydrodynamic forces are concerned, only drag forces are taken into account, since they are believed to be the predominant force acting in the resuspension process (and thus, no lift force is included in the model). Due to their small size, deposited particles are assumed to be inside the near-wall viscous sublayer. The hydrodynamic drag force acting on deposited particles is thus equal to the Stokes drag: Fdrag, || ¼ 6πFf νf Rpart Us, || f Fdrag, ^ ¼ 6πFf νf Rpart Us, ^ g
lack of data, interesting assessments of the model can be obtained by making estimations of the lacking parameters. These estimations are discussed thoroughly according to other experimental data available in the literature. The impact of different parameters (mostly related to surface features) on the numerical results is also evaluated. Influence of Surface Roughness. To investigate the impact of surface roughness on the resuspension process, numerical predictions are compared to the experimental data of Reeks et al.36 In their experiment, the resuspension of graphite and alumina particles on a polished stainless steel substrate exposed to a turbulent air flow was measured. For that purpose, particles were first allowed to settle on the stainless steel substrates. Then, the fraction of particles remaining on the surface is monitored by comparing photographs of the sample test surface after 1s exposure to a flow, whose velocity is increased step by step. Alumina particles were monodispersed 10 μm and 20 μm spheres (standard deviation around 1.16), whereas graphite particles had a mean diameter equal to 13 μm with a broader spread in size (standard deviation of 1.85). The Hamaker constants were extracted from interfacial surface energies γ available36 and using the formula from Israelachvili4 linking surface energies and Hamaker constants: AHam = γ24πz02. The Hamaker constant between alumina particles and stainless steel is first estimated to be around 1.15 1018 J, while the Hamaker constant for graphite particles interacting with stainless steel across air is assumed to be close to 3.0 1019 J. No electrostatic double layer are formed since the experiment was made in an air flow, leaving therefore only the VDW contribution in the DLVO theory. Additional parameters (such as the density or Young’s modulus) for the various substrate are given in Table 1. The associated Tabor’s parameter for sphereplate cases is equal to 1.1 with alumina particles and to 4.1 with graphite particles. The numerical procedure used is similar to the experimental one: 20 000 particles are first allowed to settle on the surface, and
ð10Þ
(Ff being the fluid density, νf the fluid kinematic viscosity, Us the velocity of the fluid seen by the particle, f = 1.7 being a correction factor accounting for the presence of the wall in the tangential direction,41 and g = 3.39 being a correction factor accounting for the presence of the wall in the normal direction42). The moment of hydrodynamic forces in the tangential direction is then given by the O’Neill formula,41 while the moment of hydrodynamic forces in the normal direction around point O is given by MO ðFdrag, || Þ ¼ 1:4Fdrag, || Rpart MO ðFdrag, ^ Þ ¼ a0 Fdrag, ^
ð11Þ
More details on the resuspension scenario are available in a previous paper on reetrainment.6 Comparison to Experimental Data. In the following part of the paper, numerical results of the resuspension model are presented and compared to experimental data. However, due to the complexity of the model (which requires several parameters such as the Hamaker constant and characteristics of the surface features), experimental data are often not comprehensive enough to allow a rigorous validation of the model. Despite this 447
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Table 1. Parameters in the Experiment of Reeks et al.36 graphite particle density (kg m3) Young’s modulus (Pa) Poissons ratio
alumina
hydrodynamic force would be required to remove particles from a surface with larger small-scale asperities. Increasing the polydispersity in small-scale asperity radius would also lead to an increased distribution of adhesion forces. Therefore, a numerical calculation has been performed using small-scale asperities of 20 nm with a standard deviation of 50 nm and a surface coverage of 1.5% (yielding the same average number of contacts), whose results compare well with experimental data, as depicted in Figure 12c (red line). Numerical predictions presented in Figure 12 are in agreement with experimental observations and have been obtained using physical parameters that seem to be relevant: a Hamaker constant of 2.5 1019 J between alumina particles and stainless steel seems more realistic than a Hamaker constant of 1.0 1018 J, while 20 nm asperities can affect the reentrainment of graphite particles. In order to study the sensitivity of the results with respect to the choice of different parameters related to surface roughness, various calculations have been performed. In the following paragraph, only the results on 10 μm alumina particles with a Hamaker constant of 1.15 1018 J are presented. First, as displayed in Figure 13a, the surface coverage of large-scale asperities influences the resuspension process of 10 μm particles: a lower surface coverage decreases the probability for particles to rock on large-scale asperities and to be removed. However, increasing the surface coverage up to 12.6% has the opposite effect: due to the high density of large-scale asperities, the probability for a particle to roll between two such asperities is small, lowering the probability of rocking. Therefore, an optimal surface coverage allows the highest resuspension rate. Second, the impact of the surface coverage of small-scale asperities has been studied: it can be seen in Figure 13b that a higher surface coverage will increase the number of particleasperity contacts, leading to higher adhesion forces (which prevent particle motion), whereas a smaller surface coverage increases the probability of having particleplate contact (which leads to tremendous adhesion forces preventing any particle motion). Thus, as was the case for the surface coverage of large-scale asperities, an optimal surface coverage of small-scale asperities provides the highest resuspension rate. Influence of Electrostatics. The influence of electrostatic conditions on the resuspension process is assessed using the experimental data of Sharma et al.11 They measured the detachment of polystyrene and glass microspheres (with a radius between 10 and 40 μm) deposited on glass substrates. Flow experiments were conducted in a rectangular flow cell, inside which particles of a desired size have been previously settled, and the release of particles from the surface was then monitored for the different flow rates. The available experimental data cover a wide range of parameters: the effect of particle diameter and of particle material as well as the effect of solution pH (and thus electrostatic conditions) were studied. The ionic strength is fixed to 0.01 M. Since the substrate is always made of glass, surface features were assumed to be independent of the case considered: no experimental data on surface roughness being available, the surface roughness is assumed to be equal to 150 nm, with a surface coverage of 0.6% (which yields satisfactory results). The Tabor’s parameter of such sphereplate systems ranges between 2.5 (which decreases to 0.4 when asperities are considered, for 40 μm polystyrene particles) and 0.27 (for 10 μm glass particles), showing that a Hamaker approach can be used.
stainless steel
2300
1600
7830
2.0 1010
3.5 1011
2.1 1011
0.3
0.3
0.29
they are then exposed to a constant flow rate during 1s. After that time, the remaining fraction of particles is extracted. No experimental data on surface features being available, surface features were assumed to be described by a two-scale distribution with parameters chosen in order to have satisfactory predictions (the influence of these parameters will be discussed later). Following Guingo and Minier,6 large-scale asperities are assumed to have a mean radius equal to 2 μm, with a surface coverage taken as 3.1%. The radius of small-scale asperities is considered to be equal to 5 nm (a preliminary evaluation of adhesion forces has shown that asperities in the range of 5 nm yield numerical predictions in agreement with experimental expectations) with a standard deviation of 50 nm (the dispersion in asperity size is assumed to follow a log-normal distribution) and a surface coverage of 0.5%. Small adhesion forces between particles and small-scale asperities allow particles to be set into motion, which can further be resuspended, upon impacting a large-scale asperity. Numerical predictions obtained using these parameters can be seen in Figure 12a,b (dashed lines): the trends for 10 and 20 μm particles are similar to the experimental ones, but the 50% critical velocity (corresponding to the friction velocity at which 50% of the particles are released) is overestimated by a factor close to 2. Numerical predictions for graphite particles are presented in Figure 12c (dashed line), showing that the dispersion in adhesion forces (and thus in reentrainment) is not as high as experimentally observed (consequently, the 50% critical velocity is underestimated). The discrepancy between experimental and numerical results for 10 and 20 μm alumina particles may stem from the evaluation of the Hamaker constant. Indeed, Maurer et al.43 determined a Hamaker constant of 23.8 1020 J for graphite, while a value of 14 1020 J is given by Israelachvili4 for alumina and (2540) 1020 J for metals such as stainless steel. Following the 11 22 1/2 4 , approximate values combining relation A12 Ham = (AHamAHam) of the Hamaker constants can be extracted: for alumina particles interacting with stainless steel, a value of 2.5 1019 J (4 times lower than in the experiment of Reeks et al.36) is obtained, whereas the Hamaker constant between graphite and stainless steel is close to 3.0 1019 J (which corresponds to the value of Reeks et al.36). The influence of the Hamaker constant on resuspension is straightforward: a higher Hamaker constant will increase the adhesion forces, making it harder to remove particles from the substrate, while a smaller Hamaker constant will reduce the adhesion forces. Using the revised estimation of the Hamaker constant for alumina particles leads to a decrease of the 50% critical velocity, and numerical results thereupon obtained are in agreement with observations (see Figure 12a,b, red lines). The discrepancy for graphite particles is rather due to a misevaluation of surface features: according to Suresh and Walz,44 the radius of surface features on a particle can be on the order 103102 particle radii. Thus, particle roughness around 20 nm or larger may have to be accounted for. Since adhesion forces increase with asperity size proportionally to the Derjaguin prefactor RpartRasp/(Rpart + Rasp), a higher 448
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Figure 12. Adhesion of alumina or graphite particles on stainless steel substrates: blue circles, experimental data;36 numerical predictions with Rlarge asp = small small 19 2 μm, Slarge J ; green dashed line, AHam = 1.0 1018 J; (b) red line, cov = 3.1%, and Rasp = 5 nm, Scov = 0.5% and with (a) red line, AHam = 2.5 10 large small small AHam = 2.5 1019 J; green dashed line, AHam = 1.0 1018 J; (c) red line, Rlarge asp = 2 μm, Scov = 3.1%, and Rasp = 20 nm, Scov = 1.5%.
Figure 13. Sensitivity of the model to the surface coverage for 10 μm alumina particles: red circle and red line, Scov; green x’s and green dashed line, Scov 4; light blue triangles and light blue dotted line, Scov 0.4.
μm with a standard deviation of (2%). Experimentally, it was shown that the critical velocity (which is defined as the velocity at which a measurable 10% release of particles is first observed) decreases with particle size. Numerical predictions have been performed using a Hamaker constant of 1.42 1020 J and a ζpotential of polystyrene particles of 2510 and 32 mV for glass
The numerical procedure is close to the one used previously: 20 000 particles are first allowed to settle on the surface, which is then exposed to a flow until a steady state is reached. Effect of Particle Diameter. First, the effect of particle diameter on the resuspension of polystyrene particles from a glass surface has been studied, using three different diameters (10, 20, and 40 449
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Figure 14. Effect of particle size on the release of polystyrene particles from a glass surface (symbols = experimental data from ref 11, lines = numerical results): red circles and red line, 10 μm particles; green x’s and green dashed line, 20 μm particles; and blue triangles and blue dotted line, 40 μm particles.
Figure 15. Effect of particle material on the release of polystyrene particles from a glass surface (symbols = experimental data from ref 11, lines = numerical results): red circles and red line, 10 μm glass particles; blue triangles and blue dashed line,10 μm polystyrene particles.
substrates.11 It can be seen in Figure 14 that the numerical predictions are in agreement with these experimental observations: the higher the particle diameter, the higher the critical velocity. One may argue that numerical predictions tend to underestimate the critical velocity of 40 μm particles, but better predictions can be obtained using a higher surface coverage (2%) or slightly larger asperities (which can be related to some surface features on the particle: as underlined by Suresh and Walz,44 the radius of surface features can be on the order of 103102 particle radii, corresponding here to radii between 20 and 200 nm). This trend is in agreement with a theoretical analysis: particle resuspension is governed by the balance between hydrodynamic forces and adhesion forces. Hydrodynamic forces are proportional to Rpart2 (the drag force is a function of Rpart and of the velocity of the fluid seen by the particle, which scales as Rpart in the near-wall region), whereas adhesion forces are proportional to Rpart. Therefore, larger particles will be easier to remove than smaller particles. Effect of Particle Material. Second, the effect of particle material has been studied using 10 μm polystyrene and glass particles deposited on a glass substrate at pH 5 (ionic strength still equal to 0.01 M). It has been found that glass particles were more easily removed from the substrate than polystyrene particles. This has been confirmed numerically, as depicted in Figure 15. As stressed by the authors, the change in the Hamaker constant (from 1.05 1020 J for the glasswaterglass system to 1.42 1020 J for the polystyrenewaterglass system) provides a partial explanation for this observation. The difference in ζ-potentials between polystyrene particles (28 mV) and glass substrates (32 mV) amplifies this trend. Effect of Solution pH. Third, the effect of pH on the resuspension of 10 μm glass particles deposited on glass substrate was studied at three different pH values (5, 7, and 10). Experimental observations underline that at higher pH conditions, glass particles are easier to remove from a glass substrate than larger particles. This is probably linked with differences in electrostatic conditions (the Hamaker constant is assumed to be constant and equals 1.05 1020 J for the glasswaterglass system): the ζpotential of glass, provided by the authors, is equal to 32 mV at pH 5 and 37 mV at pH 7 and 10. Higher ζ-potentials give rise to an increased electrostatic repulsion between surfaces and thus lower the adhesion forces. However, numerical predictions made
using these parameters show no dependence of particle removal on the electrostatic conditions. This can be due to the fact that the formulas used to calculate the electrostatic interactions were obtained using the constant-potential (CP) assumption, meaning that the surface potential remains constant when the two surfaces are brought closer one to the other, leading to a change of the surface charge. In that case, the increase of ζ-potential from 32 to 37 mV is not high enough to significantly change the adhesion forces. A numerical prediction has been made using the formula from Wiese and Healy,31 valid in the constant-charge (CC) approximation: the small values of adhesion forces in that case yield a high release of glass particles at pH 7 or 10, thus highly overestimating the resuspension rate. At pH 5, the use of the CC approximation underestimates the critical velocity. Since numerical predictions made using the CP approximation yield satisfactory results at pH 5 and since experimental observations of the resuspension rate of glass particles at pHs 7 and 10 are in between the CC and CP approximations, charge regulation appears to play a role in the determination of electrostatic forces. The classical DLVO theory considers either a CC or a CP boundary condition, but both approaches cannot reproduce the measured forces at close separation. Typically, as in the present case, the measured forces lie between the two cases of CC and CP conditions and, to improve the predictions, regulation effects have been accounted for. The notion of regulation has been introduced in previous studies4549 in order to assess the dynamics of the interaction between two double layers: upon interaction, charge and/or potential of surfaces changes due to the competition between chemical aspects (the rate of proton ad/desorption at each surface) and electrical aspects (charging of the double layer stops when electrostatic repulsion counteracts the chemical affinity of charge-determining ions for the surface). In the present paper, since we do not aim at developing a detailed charge regulation model, a simple model of charge regulation is used: this model, proposed by Carnie and Chan,50 is a linearized regulation model with a constant regulation parameter (creg) for each surface. This regulation parameter measures the competition between chemical and electrostatic aspects and PericetCamara et al.47 defined it in terms of the diffuse layer capacitance and the inner layer capacitance (both capacitance referring to various chargepotential relationships). It thus depends on the material properties and on the separation distance as well as on 450
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This adhesion model has also been coupled with a model to simulate particle resuspension from rough surfaces in a turbulent flow.6 However, as for the adhesion force model, the complexity of the resuspension model requires detailed data on surface features which are not always available in the literature. Therefore, some parameters have to be estimated, using values likely to be in the range of possible values provided by other experimental data. Numerical predictions obtained this way were quite in line with the reentrainment of particles measured by Reeks et al.36 and Sharma et al.11 The model presented in this paper is thus based on a simplified model for adhesion forces (not as detailed as a FEM method) but is still able to capture the effects of nanometric or micrometric surface roughness on the reentrainment as well as the influence of electrostatic forces on particle resuspension. Moreover, the present model allows tractable calculations of the complete deposition process (which involves hydrodynamic transport toward walls by turbulent flows, adhesion with rough surfaces, and possible reentrainment) to be performed in engineering 3D cases using only physical parameters, such as Hamaker constants, surface roughness characteristics, ζ-potentials, or ionic strength, which can be measured experimentally. The present model has been designed for cases where the adhesion between rough surfaces can be described using the DLVO theory. However, depending on the experimental conditions, the DLVO theory can fail to predict adhesion forces: for instance, it has been shown that charge regulation can alter the pH dependence of adhesion forces. Also, depending on the experimental conditions, non-DLVO forces may have to be taken into account: for example, capillary forces play a huge role in the adhesion between surfaces in high humidity conditions,51 while acidbase interactions affect the adhesion between hydrophilic or hydrophobic systems.52 Since detailed experimental data combining results on particle resuspension, adhesion forces, and surface features are scarce, it would be worth considering a more detailed approach satisfying the following requirements: a comparison of resuspension on both smooth (atomically smooth) and rough substrates, a detailed measure of surface features, and the impact of polydispersity in particle size (in relation with monodispersed particles). Calculations taking into account the competition between particle deposition and particle resuspension (and not only considering that particles are already deposited on the surface as in the present paper) have to be performed in the near future. Moreover, since particle deposition as well as particle reentrainment have been modeled quite accurately, the early stages of fouling can be simulated. This opens the way for studies of the late stage of fouling, where multilayer deposition combined with aging phenomena or deposit cohesion occurs.
Figure 16. Effect of solution pH on the release of 10 μm glass particles from a glass surface (symbols = experimental data from ref 11, lines = numerical results): red circles and red line, pH 5; green x’s and green dashed line, pH 7; and blue triangles and blue dotted line, pH 10.
pH conditions: it lies typically between 0 (the value in the CC approximation) and 1 (the value in the CP case). Within the constant-regulation CR boundary condition, each surface is described by two parameters: the surface potential and the regulation parameter (assumed to be independent of the separation distance). The advantage of the CR approximation is that the calculation is not cumbersome, while it provides an accurate description of force profile down to the contact point.47 In the present case, regulation parameters of both surfaces were chosen to be equal to 0 at pH 5 (a value of 0.4 does not affect the results), 0.98 at pH 7, and 0.995 at pH 10. Such parameters appear to be in agreement with previous theoretical works:46,47 the regulation parameter generally has a U-shaped curve in pH, where it is close to 1 (CC condition) at small and high pH values, and reaches a minimum value (more or less close to the CP condition) for intermediate values of the pH. From the results displayed in Figure 16, it can be seen that the numerical predictions obtained using such parameters are close to the experimental observations, showing a higher release of particles at higher pH conditions. From this study, it thus appears that electrostatic conditions and material properties (Hamaker constant or size) can affect particle resuspension, and the trend can be reproduced numerically using a model based on the DLVO theory accounting for charge regulation phenomena.
’ CONCLUSION In the present paper, a model to determine adhesion forces between rough surfaces has been presented. This model is based on a Hamaker approach, where surface roughness is described using hemispherical asperities. Adhesion forces and adhesion energies are calculated using the DLVO theory (i.e., the sum of van der Waals and electrostatic forces) and summing the interactions between particles and each contacting asperity. The number of asperities in contact with the particle is determined using a stochastic model with a mean value evaluated through geometric considerations. It has been shown that the numerical predictions are in agreement with several experimental data concerning both the influence of surface roughness on adhesion forces and the influence of electrostatic forces on adhesion forces. However, it should be borne in mind that numerical predictions require a detailed description of surface features and not only a characteristic average value.
’ ASSOCIATED CONTENT
bS
Supporting Information. Details about the equations giving the average number of contacts between a sphere and small-scale asperities located on large-scale asperities, details about the van der Waals and electrostatic equations used to calculate DLVO interaction energies, and tables summarizing the parameters used in the simulation for each experimental system considered. This material is available free of charge via the Internet at http://pubs.acs.org/.
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’ AUTHOR INFORMATION
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Corresponding Author
*E-mail:
[email protected] (C.H.), jean-pierre.minier@ edf.fr (J.-P.M.).
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