Article pubs.acs.org/Langmuir
Simulation of Capillary Bridges between Nanoscale Particles Michael Dörmann and Hans-Joachim Schmid* Particle Technology Group, University of Paderborn, Pohlweg 55, 33098 Paderborn, Germany ABSTRACT: Capillary forces are very important as they exceed in general other adhesion forces. But at the same time the exact calculation of these forces is very complex, so often assumptions and approximations are used. Previous research was done with regard to micrometer sized particles, but the behavior of nanoscale particles is different. Hence, the results for micrometer sized particles cannot be directly transferred when considering nanoscale particles. Therefore, a simulation method was developed to calculate numerically the shape of a rotationally symmetrical capillary bridge between two spherical particles or a particle and a plate. The capillary bridge in the gap between the particles is formed due to capillary condensation and is in thermodynamic equilibrium with the gas phase. Hence the Kelvin equation and the Young−Laplace equation can be used to calculate the profile of the capillary bridge, depending on the relative humidity of the surrounding air. The bridge profile consists of several elements that are determined consecutively and interpolated linearly. After the shape is determined, the volume and force, divided into capillary pressure force and surface tension force, can be calculated. The validation of this numerical model will be shown by comparison with several different analytical calculations for micrometer-sized particles. Furthermore, it is demonstrated that two often used approximations, (1) the toroidal approximation and (2) the use of an effective radius, cannot be used for nanoscale particles without remarkable mistake. It will be discussed how the capillary force and its components depend on different parameters, like particle size, relative humidity, contact angle, and distance, respectively. The rupture of a capillary bridge due to particle separation will also be presented. capillary bridges was done by Haines8 and Fisher.9 Later, Orr et al.10 calculated the exact solutions of capillary bridges between a sphere and a plate as elliptical integrals. With these integrals the forces of a capillary bridge and other parameters could be determined. Schubert11 calculated the forces for different configurations regarding particle shape, particle size, distance, and contact angle. The studied shapes were spheres, cones, and plates. His results were presented in dimensionless diagrams; hence, they can be used for various particle sizes and fluids. In both cases the capillary forces were determined depending on the bridge size, characterized by the so-called filling angle β (compare Figure 1). However, the relevant bridge characteristics, which includes bridge volume or Kelvin radius in the case of condensing conditions, cannot be calculated from the filling angle without knowledge of the bridge shape. Furthermore, the solutions cannot be adapted to parameter configurations that were not covered by the authors. However, these results are used to validate our simulation method for a variety of parameter sets. Other modeling approaches, that are often used, make assumptions regarding the shape of the meniscus. The exact shape of the meniscus can be described as a nodoid,10,12,13 but for an easier calculation the profile is often assumed to be a circle, so one curvature radius is constant at every point of the
1. INTRODUCTION When particles are in contact with each other and a liquid is present at the contact point, the particle interactions are dominated by capillary forces over the other adhesion forces, like van der Waals forces or electrostatic forces. On one hand the liquid can be added to the powder and forms menisci between them, and on the other hand there might be capillary condensation, when the humidity of the air condenses into the gaps between the particles. Particularly for nanoscale particles this condensation is very important. These capillary bridges induce an adhesion force due to the pressure difference between the liquid phase and the gas phase and due to the surface tension at the liquid−gas interface. Therefore, capillary bridges have a great influence on the interaction between particles and also on the behavior of bulk materials for example at agglomeration1 or powder flow2,3 as they can change the properties of the powder bulk drastically. Furthermore, they also have to be considered during AFM analyses4−6 when capillary condensation forms a meniscus between substrate and AFM tip during experiments at ambient air, and they can be even used to manipulate microscopic objects.7 As capillary forces are very important it is crucial to determine them as exact as possible. The resulting force depends on the shape of the meniscus. But the determination of the meniscus shape and hence the resulting forces is very complex, so often assumptions and approximations are used or only special cases are handled. In general, only rotationally symmetrical bridges are considered. The first research on © 2014 American Chemical Society
Received: November 14, 2013 Revised: January 9, 2014 Published: January 13, 2014 1055
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formation of liquid bridges. These studies show the influence of different parameters on the stability of the capillary bridge and with which parameter settings the capillary bridge can exist. However, the shape of the meniscus has to be presumed which was done in these studies with the toroidal approximation. Considered particle sizes are substantially larger than particles targeted in this work. To model capillary bridges between nanoscale particles a continuum mechanical approach should be used due to shorter computation time as long as the liquid phase can be treated as a continuum. This model should calculate the profile without assumptions because these were developed for micrometer sized particles. Furthermore, the approach using the Young− Laplace equation should be used if the capillary bridges are calculated depending on the relative humidity and in thermodynamic equilibrium. A model fulfilling these prerequisites will be presented here. In the next section the method and the simulation procedure will be presented. Afterward the model will be validated by several other calculations, it will be compared with two often used approximations, and the influence of several parameters will be discussed. At the end the paper will be recapitulated, and a perspective of future research will be given.
Figure 1. Schematic drawing showing the capillary bridge between (a) two spherical particles with diameter D1 and D2 and (b) a sphere with diameter D1 and a plate. The distance between the surfaces is d, θ1 and θ2 are the two contact angles, and r1 and r2 are the two curvature radii.
surface. Another approximation is to assume an effective radius, as developed by Derjaguin,14 and calculate the capillary bridge between a sphere with this effective radius and a plate instead of the bridge between two spheres with individual radii.15 These two assumptions were developed for micrometer sized particles, and therefore, one must check if they can be used also for the determination of capillary bridges between nanoscale particles. Pakarinen et al. 16 developed a method to calculate numerically the exact meniscus shape between an AFM tip and a planar surface based on the Young−Laplace equation. As only contacts to planes are considered, it is not possible to determine the forces between two particles which is of great interest to characterize bulk properties. This method will be extended here to determine the volume and the forces of capillary bridges between arbitrary sphere−sphere or sphere− plate contacts with regard to relative humidity. Another approach is to calculate the energy-minimal surface of the liquid phase and to determine the resulting force as the derivative of the energy.7 The liquid in the capillary is arranged in a way that the energy due to the interface with the particles and the gas phase will be minimized. With this approach it is even possible to model nonsymmetrical bridges, but the volume has to be fixed and the capillary force cannot be directly determined dependent on the relative humidity in thermodynamic equilibrium.7 Lambert et al.17 showed that both approaches, based on (1) the Young−Laplace equation and (2) the minimization of energy, are equivalent. Both aforementioned modeling approaches treat the capillary bridge as a continuum. In contrast to this assumption it is also possible to use molecular dynamics and calculate the interactions between all molecules.4,18 Here the molecular properties of the materials have to be known in contrast to macroscopic properties in continuum mechanical approaches. The computational costs are very high, so that only very small particles with diameters up to about 4 nm can be calculated within reasonable time. Furthermore, this simulation method is not deterministic, so the results show stochastic fluctuations, and they have to be evaluated statistically. An interesting alternative approach to liquid bridges is the thermodynamic stability analysis as done by Elliott.19,20 This analysis can be used to get information on the stability and the
2. METHODS The modeling approach presented in this paper calculates the profile of a capillary bridge between two spherical particles (Figure 1a) or between a spherical particle and a plate (Figure 1b) in the case of condensing conditions. The capillary bridge and hence the particles have to be rotationally symmetrical. The relevant parameters in the case of spherical particles are the particle diameters, D1 and D2, the contact angles, θ1 and θ2, the distance d, and the relative humidity φ. All geometrical parameters can be chosen independently. The relative humidity can be set to a value between 0% and 100%. The capillary bridge is formed in the gap between these two objects and can be described by two curvature radii, r1 and r2. These curvature radii can be combined to a mean curvature radius rK with −1 ⎛1 1⎞ rK = ⎜ + ⎟ r2 ⎠ ⎝ r1
(1)
The mean curvature radius is constant at every point on the surface under consideration of two assumptions:21 (1) The liquid phase is in thermodynamic equilibrium with the surrounding air, and (2) gravity can be neglected. The second assumption is valid for sufficiently small capillary bridges.22 Then, the mean curvature radius can be determined with the Kelvin equation
rK =
γ ·VM ⎛ p⎞ RT ln⎜ p ⎟ ⎝ s⎠
(2)
and it is also called Kelvin radius.23 Here, γ is the surface tension of the liquid, VM is the molar volume of the liquid phase, R is the gas constant, T is the temperature, p is the vapor pressure, and ps is the equilibrium vapor pressure of the plane surface. In the case of water as liquid phase and air as continuous phase the quotient p/ps is equivalent to the relative humidity of the air. Due to the curved surface there exists a pressure difference Δp between the meniscus and the surrounding air. This pressure difference is also called Laplace pressure. It can be calculated by the Young−Laplace equation23
Δp =
γ rK
(3)
with the Kelvin radius and the surface tension. 1056
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The capillary bridge between the two objects induces an adhesion force FC that can be divided into two components as
FC = FP + FS
When the shape of the capillary bridge is calculated correctly, the adhesion force caused by the bridge can be determined according to eq 5. The force could be calculated at an arbitrary cross-section of the liquid bridge. The total force would remain constant, but the ratio of the two components would change. In this paper the forces are calculated at the neck of the meniscus. The volume of the capillary bridge is determined with the following method. The volume of revolution between all points is calculated and summed up. At the end the volumes of the two spherical caps of the particles that reach into the liquid phase are subtracted. As another convergence criterion the resulting force and volume have to agree with the results of the iteration before; otherwise, the starting point will be varied.
(4)
The first component, FP, is due to the pressure difference between meniscus and air, and the second component, FS, originates from the surface tension on the interface between fluid phase and gas phase. If the profile of the meniscus is known, the capillary force in normal direction can be calculated at any cross-section of the liquid bridge with
FC = Δpπr 2 + 2γπr cos(α)
(5)
The first term on the right-hand side represents again the capillary pressure force and the second term the surface tension force. The radius of the cross-section of the capillary bridge is denoted with r, and α is the angle the interface deviates from the normal direction in this cross-section (see Figure 1a). The shape of the capillary bridge can be calculated numerically with the following modeling approach. The meniscus is modeled to consist of many distinct points, and two neighboring points are connected by a straight line. The interval between these points is set constant. In the following, the iterative procedure is presented. A filling angle β of the first particle is estimated, and the corresponding point on the surface of the first particle is calculated. The gradient of the meniscus is determined by the curvature of the sphere and the contact angle. With the gradient and the point-to-point interval the next point can be calculated. Because the capillary bridge is rotationally symmetrical, the center of the circle of the radius r2 is on the rotation axis of the capillary bridge.11 Therefore, the curvature radius r2 can be calculated as distance from the rotation axis, and afterward, r1 can be calculated with eq 1 under consideration of the Kelvin radius. For rotationally symmetrical bridges the profile of the meniscus y(x) is linked to the curvature radius r1 by
r1 =
⎡ ⎢⎣1 +
3. RESULTS AND DISCUSSION In the next section the simulation will be validated by several different methods that also calculate the exact capillary force. Afterward the results are compared with often used approximations, and the influence of different parameters is shown. In all simulations water as the liquid phase and air as the gaseous phase are assumed. The temperature of the system is set to 20 °C. 3.1. Validation. In a first step the results provided by this simulation were validated with the exact calculated results by Orr et al.10 and by Schubert.11 Furthermore, the results were compared with the simulation of Pakarinen et al.16 Figure 3a shows the very good agreement between the simulation presented in this paper and the calculation by Orr et al. for a sphere−plate configuration. Only for a contact angle of 40° and high filling angles can a difference between both
3/2 dy 2 ⎤ dx
( ) ⎥⎦ d2y dx 2
(6)
With this equation it is possible to calculate the gradient of the meniscus at this point and hence the position of the next point. This procedure is continued until the meniscus reaches the second particle. The contact angle on the second particle is checked to match the desired contact angle within numerical precision. If the contact angle does not match the desired contact angle or the meniscus does not reach the second particle at all, the filling angle on the first particle and, therefore, the starting point will be varied. This iteration method is shown in Figure 2.
Figure 2. Iteration sequence.
The maximum filling angle which is considered is 90°. For a greater filling angle it is possible that the calculated meniscus intersects the particle surface leading to incorrect results. Normally the filling angle is smaller than 90°. Only in extreme cases, for example contact between a very small particle and a plate at a relative humidity near to saturation, can the filling angle exceed 90°.
Figure 3. Comparison of the capillary force calculated with this work with the results (a) by Orr et al.10 and (b) by Schubert11 and by Pakarinen et al.16 1057
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methods be seen. But even the difference at a filling angle of 60° is less than 1%, and these filling angles could only be obtained at a humidity near saturation and very small particle sizes. A comparison of the results by Schubert for various sphere−sphere and sphere−plate configurations and by Pakarinen et al. for different sphere−plate configurations with capillary forces calculated with this work is shown in Figure 3b, and the deviations are under 5% which is a very good agreement considering reading accuracy. Hence, classical, macroscopic solutions show very good agreement with this work. 3.2. Comparison with Often Used Assumptions. To reduce the calculation complexity, approximations and simplifications are often used. The most used assumption is that the profile of the capillary bridge can be described as a circle, and hence, the curvature radius r1 remains constant for every point of the meniscus profile. Then the capillary force can be calculated depending on the filling angle analytically. The forces between two spherical particles were calculated with this approximation with equations given by Butt and Kappl24 and compared to the solutions of this work. The difference in the total capillary force between both methods is shown in Figure 4. For relatively big particles in direct contact, in these cases with a diameter of 500 nm or more, the difference between both methods is small, here under 5%. But if the particle size decreases or the distance increases only slightly the difference grows especially at relative humidities near the minimum humidity for capillary bridge formation. In these examples the difference might become larger than 40%. It should be noted that the approximation originally was developed for capillary bridges between micrometer sized particles where one curvature radius is much greater than the other one. In this case this assumption is reasonable. But for nanoscale particles both curvature radii are at about the same size. Thus, it is not possible that both, the Kelvin radius and one curvature radius, can be constant while the second curvature radius varies. The deviation increases with a greater distance as the length of the capillary bridge increases, and hence, the difference between the real profile and an assumed circular profile is larger. Furthermore, this work predicts the existence of capillary bridges at lower relative humidities compared to the simplified calculation assuming a circular meniscus shape. The gradient of the profile is steeper when it is approximated as a circle than when it is a nodoid. Due to this the capillary bridge calculated with this work can cover a longer distance assuming the same curvature. The distance of 0 nm was chosen because it is used very often. The distance of 0.4 nm is the classical distance for calculations of the van der Waals force. Despite the open question, which is the correct minimum contact distance, Figure 4 shows clearly that the circular approximation should not be used for nanoscale particles. A second often used assumption is the use of an effective radius or diameter. In this case the capillary bridge between two spherical particles with the diameters D 1 and D 2 is approximated as a capillary bridge between a particle with an effective diameter D* and a plate. The effective diameter D* is determined depending on the two original diameters with the relation D1D2 D* = D1 + D2 (7)
Figure 4. Relative difference between capillary forces calculated with circular approximation Fcirc.appr. and with this work Fthis_work between two spheres with a contact angle Θ of 0° and different particle size D at distance (a) d = 0 nm, (b) d = 0.2 nm, and (c) d = 0.4 nm.
general easier than the calculation of the meniscus between two spheres. Figure 5 shows the relative differences of the force and the volume of the capillary bridge between the calculations with the effective radius and with this work, respectively. Simulations, in which the filling angle exceeds 90°, are not considered here, which only occurs for simulations between a small sphere and a plate at a high relative humidity. Near to saturation and for smaller particle sizes the difference between the calculated forces increases. Furthermore, the shape of the meniscus is different because it is determined between different objects. Due to this also the distribution into the two force components differs between these two approaches. This assumption has therefore several disadvantages to this work as the filling angle can exceed 90°, leading to calculation errors,
The advantage is that the calculation of the capillary force between a sphere and a plate has to be handled, which is in 1058
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capillary force depending on particle size is shown. The force decreases with smaller particle size as the size of the capillary bridge is also decreasing. For particles with a diameter over 1 μm the force scales nearly linearly with the particle diameter. But for smaller particles the decrease of the force is higher than the size decrease because the influence of the particle curvature becomes stronger. With increasing relative humidity the capillary force first increases until a plateau is reached because the bridge gets bigger and the perimeter and area of the cross-section increase, too. After this plateau the total force decreases despite an increasing bridge volume due to more water condensing in the meniscus. With higher relative humidity the Kelvin radius increases and the capillary pressure decreases. This decrease of the capillary pressure has a negative effect on the force that cannot be compensated by the bigger size of the meniscus. Another important parameter is the contact angle of the surface. The resulting forces for several contact angles are shown in Figure 7. As expected, when the surface becomes more hydrophobic, that is, the contact angle increases, the resulting force decreases. From evaluation of the curves in Figure 7, it appears that the capillary force is approximately proportional to the cosine of the contact angle. The decreasing force is caused by smaller filling angles and smaller bridge
Figure 5. Relative difference of (a) force and (b) volume of a capillary bridge calculated with effective diameter (Feff.diam., respectively, Veff.diam.) and with this work (Fthis_work, respectively, Vthis_work) between two spheres with contact angle Θ of 0° and different particle size D at a distance d of 0 nm.
and the shape of the meniscus is different, and hence also volume and force. 3.3. Influence of Different Parameters. The capillary force is dependent on several different parameters. The most obvious parameter is the particle size. In Figure 6 the total
Figure 7. Capillary force FC depending on relative humidity φ and contact angle Θ for two equal sized particles with a diameter D of (a) 100 nm and (b) 10 nm at a distance d of 0.2 nm and a surface tension of the liquid γ of 72.74 mN/m.
Figure 6. Dimensionless force depending on relative humidity φ and particle size D between two equal sized particles with a contact angle Θ of 0° at a distance d of 0.2 nm. 1059
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volumes. Furthermore, no capillary bridges can be predicted at higher contact angles and small relative humidities because the maximum length of the meniscus is decreasing due to the higher contact angles. Therefore, no thermodynamically stable capillary bridges should exist between particles in these cases. Also, the distance between the particles is influencing the shape and the force of the capillary bridge. The dependence of the capillary force on the separation distance at different relative humidites is displayed in Figure 8. At direct contact the
Figure 9. Dependence of capillary force on distance d calculated with constant volume or in thermodynamic equilibrium for capillary bridges between a sphere with a diameter D of 100 nm and a plate, both with a contact angle Θ of 30° at a relative humidity φ of 0.95 and a surface tension of the liquid γ of 72.74 mN/m.
surface are in direct contact under consideration of thermodynamic equilibrium. The calculated volume was set constant, and the shape of the capillary bridge was calculated at different distances. These simulations were performed with the public domain software Surface Evolver.25 It allows the determination of the energy-minimizing surface of a meniscus with a fixed volume. For the other possibility the simulation presented in this paper was performed with constant parameters at increasing distances until no meniscus could be calculated anymore. While at small distances the difference between both assumptions is very small, this increases as the distance increases. When the system has to be in equilibrium, water evaporates, and therefore, the volume decreases at higher distances. The difference in the volume is greater for larger separation distance leading to greater difference regarding the capillary force. It has also a very substantial influence on the maximum distance when the capillary bridge breaks. In the case presented here, this maximum distance for the constant volume is about 4 times higher than that for the modeling in thermodynamic equilibrium. Another important piece of information is how the capillary force is divided into capillary pressure force and surface tension force. In general the force based on the capillary pressure is considered to be dominant over the surface tension force.23 This is the case for micrometer sized particles and the respective capillary bridges. The surface tension can even be neglected without a substantial mistake. But as the particles become smaller, the surface tension force becomes more important, which can be seen in Figure 10. The capillary bridges between smaller particles are in general smaller than between bigger particles. If the cross-section of the bridge is smaller, the perimeter, and thus the surface tension force, becomes more important compared to the cross-sectional area, and thus the capillary pressure force. Furthermore, with increasing relative humidity the ratio of the capillary pressure force declines as the capillary pressure also decreases with growing relative humidity and Kelvin radius.
Figure 8. Total capillary force FC depending on separation distance d and relative humidity φ for capillary bridges between two spheres with a diameter D of (a) 100 nm and (b) 10 nm and a contact angle Θ of 0° and a surface tension of the liquid γ of 72.74 mN/m at equilibrium.
capillary force is at a maximum, and it decreases steadily while the particles are separated from each other. At a certain distance the capillary bridge ruptures. This rupture distance increases with higher relative humidity as more water condenses in the meniscus. At high relative humidities the capillary force is not as dependent on the distance as at lower humidities. Due to the higher capillary pressure, the force at direct contact is slightly higher at lower humidities as was mentioned before. When separating two particles, two different cases have to be considered: (1) The capillary bridge is during the whole process in equilibrium with the surrounding air, or (2) the volume of the fluid phase remains constant. In the case of a particle pull-off from a plate, the results of these two possibilities differ very much, as can be seen in Figure 9. First the force and the volume were calculated when particle and
4. CONCLUSION In this paper a simulation was presented that calculates numerically the exact shape as well as resulting forces of a capillary bridge in the case of capillary condensation, that means it is dependent on the relative humidity of the air. 1060
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which minimum particle size the continuum assumption for the liquid bridge is still valid. Additionally, this comparison shall reveal the behavior at small relative humidities since there has to be a minimum amount of water on the surface that it behaves like a liquid phase26 and also the curvature of the meniscus may reach values in the range of molecule sizes.27 Furthermore, the simulation will be enhanced to determine also nonrotational-symmetrical bridges and the transient behavior of the meniscus. This allows the calculation of capillary bridges between particles that are not rotational-symmetrical or that have arbitrary roughnesses. Also, the forces that do not act in normal direction can be calculated, and the dynamics of particle separation in normal and non-normal direction can be determined.
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Figure 10. Ratio of capillary pressure force FP on total force FC for different particles sizes D and relative humidities φ between two equal sized spheres with a distance d of 0.2 nm.
AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected].
Particular attention has been paid to nanoscale particles, since capillary condensation is much more important in that case. To validate this model it was shown that results obtained with this simulation agree very well with several other analytical methods to calculate the capillary force. Furthermore, two approximations that are often used were compared to this work. The first approximation is to consider the meniscus profile as a circle to simplify the calculation. This assumption shows good agreement for micrometer sized particles, but as the particles become smaller the deviation to this work increases strongly since the implied assumptions are not valid anymore at this scale. Furthermore, this work predicts the existence of capillary bridges at lower humidities compared to simulations using this approximation. The second popularly used assumption is the use of an effective radius or diameter that allows one to calculate the capillary bridge between a sphere and a plate instead of the capillary bridge between two spheres. It is shown that the results differ for small particle sizes at high relative humidities only, but the shape of the capillary bridge is different, and due to the higher filling angle numerical problems can occur. Therefore, this work gives more accurate results especially for nanoscale particles and should be preferred. It was shown that capillary bridges depend on several different parameters. As the particle size increases also the resulting force increases. But the distribution in the components changes. For micrometer sized particles the capillary pressure force is dominant, whereas surface tension force becomes increasingly important for decreasing particle sizes. A higher relative humidity leads to bigger capillary bridges, suggesting a higher capillary force, but at the same time the capillary pressure decreases. Hence there is a maximum of the resulting force with respect to relative humidity. A higher contact angle, that is, a more hydrophobic surface, decreases the capillary forces, and also a higher relative humidity is necessary for the formation of a meniscus. When the particles are separated from each other the adhesion force decreases quite slowly first until no capillary bridge can exist anymore and it breaks. A comparison between two possibilities of modeling the particle separation was shown, that is, assuming the capillary bridge in thermodynamic equilibrium and assuming a constant bridge volume, respectively. It is shown that a constant bridge volume leads to the existence of a liquid bridge up to much larger distances compared to the assumption of equilibrium. In the future a comparison to results from molecular dynamics simulations will be made in order to check down to
Author Contributions
The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thank Deutsche Forschungsgemeinschaft (DFG) for the financial funding of this project in the priority program “Partikel im Kontakt−Mikromechanik, Mikroprozessdynamik und Partikelkollektive” (SPP 1486) under grant SCHM1429/9.
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Simulation of Capillary Bridges between Nanoscale Particles Michael Dörmann and Hans-Joachim Schmid* Particle Technology Group, University of Paderborn, Pohlweg 55, 33098 Paderborn, Germany ABSTRACT: Capillary forces are very important as they exceed in general other adhesion forces. But at the same time the exact calculation of these forces is very complex, so often assumptions and approximations are used. Previous research was done with regard to micrometer sized particles, but the behavior of nanoscale particles is different. Hence, the results for micrometer sized particles cannot be directly transferred when considering nanoscale particles. Therefore, a simulation method was developed to calculate numerically the shape of a rotationally symmetrical capillary bridge between two spherical particles or a particle and a plate. The capillary bridge in the gap between the particles is formed due to capillary condensation and is in thermodynamic equilibrium with the gas phase. Hence the Kelvin equation and the Young−Laplace equation can be used to calculate the profile of the capillary bridge, depending on the relative humidity of the surrounding air. The bridge profile consists of several elements that are determined consecutively and interpolated linearly. After the shape is determined, the volume and force, divided into capillary pressure force and surface tension force, can be calculated. The validation of this numerical model will be shown by comparison with several different analytical calculations for micrometer-sized particles. Furthermore, it is demonstrated that two often used approximations, (1) the toroidal approximation and (2) the use of an effective radius, cannot be used for nanoscale particles without remarkable mistake. It will be discussed how the capillary force and its components depend on different parameters, like particle size, relative humidity, contact angle, and distance, respectively. The rupture of a capillary bridge due to particle separation will also be presented. capillary bridges was done by Haines8 and Fisher.9 Later, Orr et al.10 calculated the exact solutions of capillary bridges between a sphere and a plate as elliptical integrals. With these integrals the forces of a capillary bridge and other parameters could be determined. Schubert11 calculated the forces for different configurations regarding particle shape, particle size, distance, and contact angle. The studied shapes were spheres, cones, and plates. His results were presented in dimensionless diagrams; hence, they can be used for various particle sizes and fluids. In both cases the capillary forces were determined depending on the bridge size, characterized by the so-called filling angle β (compare Figure 1). However, the relevant bridge characteristics, which includes bridge volume or Kelvin radius in the case of condensing conditions, cannot be calculated from the filling angle without knowledge of the bridge shape. Furthermore, the solutions cannot be adapted to parameter configurations that were not covered by the authors. However, these results are used to validate our simulation method for a variety of parameter sets. Other modeling approaches, that are often used, make assumptions regarding the shape of the meniscus. The exact shape of the meniscus can be described as a nodoid,10,12,13 but for an easier calculation the profile is often assumed to be a circle, so one curvature radius is constant at every point of the
1. INTRODUCTION When particles are in contact with each other and a liquid is present at the contact point, the particle interactions are dominated by capillary forces over the other adhesion forces, like van der Waals forces or electrostatic forces. On one hand the liquid can be added to the powder and forms menisci between them, and on the other hand there might be capillary condensation, when the humidity of the air condenses into the gaps between the particles. Particularly for nanoscale particles this condensation is very important. These capillary bridges induce an adhesion force due to the pressure difference between the liquid phase and the gas phase and due to the surface tension at the liquid−gas interface. Therefore, capillary bridges have a great influence on the interaction between particles and also on the behavior of bulk materials for example at agglomeration1 or powder flow2,3 as they can change the properties of the powder bulk drastically. Furthermore, they also have to be considered during AFM analyses4−6 when capillary condensation forms a meniscus between substrate and AFM tip during experiments at ambient air, and they can be even used to manipulate microscopic objects.7 As capillary forces are very important it is crucial to determine them as exact as possible. The resulting force depends on the shape of the meniscus. But the determination of the meniscus shape and hence the resulting forces is very complex, so often assumptions and approximations are used or only special cases are handled. In general, only rotationally symmetrical bridges are considered. The first research on © 2014 American Chemical Society
Received: November 14, 2013 Revised: January 9, 2014 Published: January 13, 2014 1055
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formation of liquid bridges. These studies show the influence of different parameters on the stability of the capillary bridge and with which parameter settings the capillary bridge can exist. However, the shape of the meniscus has to be presumed which was done in these studies with the toroidal approximation. Considered particle sizes are substantially larger than particles targeted in this work. To model capillary bridges between nanoscale particles a continuum mechanical approach should be used due to shorter computation time as long as the liquid phase can be treated as a continuum. This model should calculate the profile without assumptions because these were developed for micrometer sized particles. Furthermore, the approach using the Young− Laplace equation should be used if the capillary bridges are calculated depending on the relative humidity and in thermodynamic equilibrium. A model fulfilling these prerequisites will be presented here. In the next section the method and the simulation procedure will be presented. Afterward the model will be validated by several other calculations, it will be compared with two often used approximations, and the influence of several parameters will be discussed. At the end the paper will be recapitulated, and a perspective of future research will be given.
Figure 1. Schematic drawing showing the capillary bridge between (a) two spherical particles with diameter D1 and D2 and (b) a sphere with diameter D1 and a plate. The distance between the surfaces is d, θ1 and θ2 are the two contact angles, and r1 and r2 are the two curvature radii.
surface. Another approximation is to assume an effective radius, as developed by Derjaguin,14 and calculate the capillary bridge between a sphere with this effective radius and a plate instead of the bridge between two spheres with individual radii.15 These two assumptions were developed for micrometer sized particles, and therefore, one must check if they can be used also for the determination of capillary bridges between nanoscale particles. Pakarinen et al. 16 developed a method to calculate numerically the exact meniscus shape between an AFM tip and a planar surface based on the Young−Laplace equation. As only contacts to planes are considered, it is not possible to determine the forces between two particles which is of great interest to characterize bulk properties. This method will be extended here to determine the volume and the forces of capillary bridges between arbitrary sphere−sphere or sphere− plate contacts with regard to relative humidity. Another approach is to calculate the energy-minimal surface of the liquid phase and to determine the resulting force as the derivative of the energy.7 The liquid in the capillary is arranged in a way that the energy due to the interface with the particles and the gas phase will be minimized. With this approach it is even possible to model nonsymmetrical bridges, but the volume has to be fixed and the capillary force cannot be directly determined dependent on the relative humidity in thermodynamic equilibrium.7 Lambert et al.17 showed that both approaches, based on (1) the Young−Laplace equation and (2) the minimization of energy, are equivalent. Both aforementioned modeling approaches treat the capillary bridge as a continuum. In contrast to this assumption it is also possible to use molecular dynamics and calculate the interactions between all molecules.4,18 Here the molecular properties of the materials have to be known in contrast to macroscopic properties in continuum mechanical approaches. The computational costs are very high, so that only very small particles with diameters up to about 4 nm can be calculated within reasonable time. Furthermore, this simulation method is not deterministic, so the results show stochastic fluctuations, and they have to be evaluated statistically. An interesting alternative approach to liquid bridges is the thermodynamic stability analysis as done by Elliott.19,20 This analysis can be used to get information on the stability and the
2. METHODS The modeling approach presented in this paper calculates the profile of a capillary bridge between two spherical particles (Figure 1a) or between a spherical particle and a plate (Figure 1b) in the case of condensing conditions. The capillary bridge and hence the particles have to be rotationally symmetrical. The relevant parameters in the case of spherical particles are the particle diameters, D1 and D2, the contact angles, θ1 and θ2, the distance d, and the relative humidity φ. All geometrical parameters can be chosen independently. The relative humidity can be set to a value between 0% and 100%. The capillary bridge is formed in the gap between these two objects and can be described by two curvature radii, r1 and r2. These curvature radii can be combined to a mean curvature radius rK with −1 ⎛1 1⎞ rK = ⎜ + ⎟ r2 ⎠ ⎝ r1
(1)
The mean curvature radius is constant at every point on the surface under consideration of two assumptions:21 (1) The liquid phase is in thermodynamic equilibrium with the surrounding air, and (2) gravity can be neglected. The second assumption is valid for sufficiently small capillary bridges.22 Then, the mean curvature radius can be determined with the Kelvin equation
rK =
γ ·VM ⎛ p⎞ RT ln⎜ p ⎟ ⎝ s⎠
(2)
and it is also called Kelvin radius.23 Here, γ is the surface tension of the liquid, VM is the molar volume of the liquid phase, R is the gas constant, T is the temperature, p is the vapor pressure, and ps is the equilibrium vapor pressure of the plane surface. In the case of water as liquid phase and air as continuous phase the quotient p/ps is equivalent to the relative humidity of the air. Due to the curved surface there exists a pressure difference Δp between the meniscus and the surrounding air. This pressure difference is also called Laplace pressure. It can be calculated by the Young−Laplace equation23
Δp =
γ rK
(3)
with the Kelvin radius and the surface tension. 1056
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The capillary bridge between the two objects induces an adhesion force FC that can be divided into two components as
FC = FP + FS
When the shape of the capillary bridge is calculated correctly, the adhesion force caused by the bridge can be determined according to eq 5. The force could be calculated at an arbitrary cross-section of the liquid bridge. The total force would remain constant, but the ratio of the two components would change. In this paper the forces are calculated at the neck of the meniscus. The volume of the capillary bridge is determined with the following method. The volume of revolution between all points is calculated and summed up. At the end the volumes of the two spherical caps of the particles that reach into the liquid phase are subtracted. As another convergence criterion the resulting force and volume have to agree with the results of the iteration before; otherwise, the starting point will be varied.
(4)
The first component, FP, is due to the pressure difference between meniscus and air, and the second component, FS, originates from the surface tension on the interface between fluid phase and gas phase. If the profile of the meniscus is known, the capillary force in normal direction can be calculated at any cross-section of the liquid bridge with
FC = Δpπr 2 + 2γπr cos(α)
(5)
The first term on the right-hand side represents again the capillary pressure force and the second term the surface tension force. The radius of the cross-section of the capillary bridge is denoted with r, and α is the angle the interface deviates from the normal direction in this cross-section (see Figure 1a). The shape of the capillary bridge can be calculated numerically with the following modeling approach. The meniscus is modeled to consist of many distinct points, and two neighboring points are connected by a straight line. The interval between these points is set constant. In the following, the iterative procedure is presented. A filling angle β of the first particle is estimated, and the corresponding point on the surface of the first particle is calculated. The gradient of the meniscus is determined by the curvature of the sphere and the contact angle. With the gradient and the point-to-point interval the next point can be calculated. Because the capillary bridge is rotationally symmetrical, the center of the circle of the radius r2 is on the rotation axis of the capillary bridge.11 Therefore, the curvature radius r2 can be calculated as distance from the rotation axis, and afterward, r1 can be calculated with eq 1 under consideration of the Kelvin radius. For rotationally symmetrical bridges the profile of the meniscus y(x) is linked to the curvature radius r1 by
r1 =
⎡ ⎢⎣1 +
3. RESULTS AND DISCUSSION In the next section the simulation will be validated by several different methods that also calculate the exact capillary force. Afterward the results are compared with often used approximations, and the influence of different parameters is shown. In all simulations water as the liquid phase and air as the gaseous phase are assumed. The temperature of the system is set to 20 °C. 3.1. Validation. In a first step the results provided by this simulation were validated with the exact calculated results by Orr et al.10 and by Schubert.11 Furthermore, the results were compared with the simulation of Pakarinen et al.16 Figure 3a shows the very good agreement between the simulation presented in this paper and the calculation by Orr et al. for a sphere−plate configuration. Only for a contact angle of 40° and high filling angles can a difference between both
3/2 dy 2 ⎤ dx
( ) ⎥⎦ d2y dx 2
(6)
With this equation it is possible to calculate the gradient of the meniscus at this point and hence the position of the next point. This procedure is continued until the meniscus reaches the second particle. The contact angle on the second particle is checked to match the desired contact angle within numerical precision. If the contact angle does not match the desired contact angle or the meniscus does not reach the second particle at all, the filling angle on the first particle and, therefore, the starting point will be varied. This iteration method is shown in Figure 2.
Figure 2. Iteration sequence.
The maximum filling angle which is considered is 90°. For a greater filling angle it is possible that the calculated meniscus intersects the particle surface leading to incorrect results. Normally the filling angle is smaller than 90°. Only in extreme cases, for example contact between a very small particle and a plate at a relative humidity near to saturation, can the filling angle exceed 90°.
Figure 3. Comparison of the capillary force calculated with this work with the results (a) by Orr et al.10 and (b) by Schubert11 and by Pakarinen et al.16 1057
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methods be seen. But even the difference at a filling angle of 60° is less than 1%, and these filling angles could only be obtained at a humidity near saturation and very small particle sizes. A comparison of the results by Schubert for various sphere−sphere and sphere−plate configurations and by Pakarinen et al. for different sphere−plate configurations with capillary forces calculated with this work is shown in Figure 3b, and the deviations are under 5% which is a very good agreement considering reading accuracy. Hence, classical, macroscopic solutions show very good agreement with this work. 3.2. Comparison with Often Used Assumptions. To reduce the calculation complexity, approximations and simplifications are often used. The most used assumption is that the profile of the capillary bridge can be described as a circle, and hence, the curvature radius r1 remains constant for every point of the meniscus profile. Then the capillary force can be calculated depending on the filling angle analytically. The forces between two spherical particles were calculated with this approximation with equations given by Butt and Kappl24 and compared to the solutions of this work. The difference in the total capillary force between both methods is shown in Figure 4. For relatively big particles in direct contact, in these cases with a diameter of 500 nm or more, the difference between both methods is small, here under 5%. But if the particle size decreases or the distance increases only slightly the difference grows especially at relative humidities near the minimum humidity for capillary bridge formation. In these examples the difference might become larger than 40%. It should be noted that the approximation originally was developed for capillary bridges between micrometer sized particles where one curvature radius is much greater than the other one. In this case this assumption is reasonable. But for nanoscale particles both curvature radii are at about the same size. Thus, it is not possible that both, the Kelvin radius and one curvature radius, can be constant while the second curvature radius varies. The deviation increases with a greater distance as the length of the capillary bridge increases, and hence, the difference between the real profile and an assumed circular profile is larger. Furthermore, this work predicts the existence of capillary bridges at lower relative humidities compared to the simplified calculation assuming a circular meniscus shape. The gradient of the profile is steeper when it is approximated as a circle than when it is a nodoid. Due to this the capillary bridge calculated with this work can cover a longer distance assuming the same curvature. The distance of 0 nm was chosen because it is used very often. The distance of 0.4 nm is the classical distance for calculations of the van der Waals force. Despite the open question, which is the correct minimum contact distance, Figure 4 shows clearly that the circular approximation should not be used for nanoscale particles. A second often used assumption is the use of an effective radius or diameter. In this case the capillary bridge between two spherical particles with the diameters D 1 and D 2 is approximated as a capillary bridge between a particle with an effective diameter D* and a plate. The effective diameter D* is determined depending on the two original diameters with the relation D1D2 D* = D1 + D2 (7)
Figure 4. Relative difference between capillary forces calculated with circular approximation Fcirc.appr. and with this work Fthis_work between two spheres with a contact angle Θ of 0° and different particle size D at distance (a) d = 0 nm, (b) d = 0.2 nm, and (c) d = 0.4 nm.
general easier than the calculation of the meniscus between two spheres. Figure 5 shows the relative differences of the force and the volume of the capillary bridge between the calculations with the effective radius and with this work, respectively. Simulations, in which the filling angle exceeds 90°, are not considered here, which only occurs for simulations between a small sphere and a plate at a high relative humidity. Near to saturation and for smaller particle sizes the difference between the calculated forces increases. Furthermore, the shape of the meniscus is different because it is determined between different objects. Due to this also the distribution into the two force components differs between these two approaches. This assumption has therefore several disadvantages to this work as the filling angle can exceed 90°, leading to calculation errors,
The advantage is that the calculation of the capillary force between a sphere and a plate has to be handled, which is in 1058
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capillary force depending on particle size is shown. The force decreases with smaller particle size as the size of the capillary bridge is also decreasing. For particles with a diameter over 1 μm the force scales nearly linearly with the particle diameter. But for smaller particles the decrease of the force is higher than the size decrease because the influence of the particle curvature becomes stronger. With increasing relative humidity the capillary force first increases until a plateau is reached because the bridge gets bigger and the perimeter and area of the cross-section increase, too. After this plateau the total force decreases despite an increasing bridge volume due to more water condensing in the meniscus. With higher relative humidity the Kelvin radius increases and the capillary pressure decreases. This decrease of the capillary pressure has a negative effect on the force that cannot be compensated by the bigger size of the meniscus. Another important parameter is the contact angle of the surface. The resulting forces for several contact angles are shown in Figure 7. As expected, when the surface becomes more hydrophobic, that is, the contact angle increases, the resulting force decreases. From evaluation of the curves in Figure 7, it appears that the capillary force is approximately proportional to the cosine of the contact angle. The decreasing force is caused by smaller filling angles and smaller bridge
Figure 5. Relative difference of (a) force and (b) volume of a capillary bridge calculated with effective diameter (Feff.diam., respectively, Veff.diam.) and with this work (Fthis_work, respectively, Vthis_work) between two spheres with contact angle Θ of 0° and different particle size D at a distance d of 0 nm.
and the shape of the meniscus is different, and hence also volume and force. 3.3. Influence of Different Parameters. The capillary force is dependent on several different parameters. The most obvious parameter is the particle size. In Figure 6 the total
Figure 7. Capillary force FC depending on relative humidity φ and contact angle Θ for two equal sized particles with a diameter D of (a) 100 nm and (b) 10 nm at a distance d of 0.2 nm and a surface tension of the liquid γ of 72.74 mN/m.
Figure 6. Dimensionless force depending on relative humidity φ and particle size D between two equal sized particles with a contact angle Θ of 0° at a distance d of 0.2 nm. 1059
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volumes. Furthermore, no capillary bridges can be predicted at higher contact angles and small relative humidities because the maximum length of the meniscus is decreasing due to the higher contact angles. Therefore, no thermodynamically stable capillary bridges should exist between particles in these cases. Also, the distance between the particles is influencing the shape and the force of the capillary bridge. The dependence of the capillary force on the separation distance at different relative humidites is displayed in Figure 8. At direct contact the
Figure 9. Dependence of capillary force on distance d calculated with constant volume or in thermodynamic equilibrium for capillary bridges between a sphere with a diameter D of 100 nm and a plate, both with a contact angle Θ of 30° at a relative humidity φ of 0.95 and a surface tension of the liquid γ of 72.74 mN/m.
surface are in direct contact under consideration of thermodynamic equilibrium. The calculated volume was set constant, and the shape of the capillary bridge was calculated at different distances. These simulations were performed with the public domain software Surface Evolver.25 It allows the determination of the energy-minimizing surface of a meniscus with a fixed volume. For the other possibility the simulation presented in this paper was performed with constant parameters at increasing distances until no meniscus could be calculated anymore. While at small distances the difference between both assumptions is very small, this increases as the distance increases. When the system has to be in equilibrium, water evaporates, and therefore, the volume decreases at higher distances. The difference in the volume is greater for larger separation distance leading to greater difference regarding the capillary force. It has also a very substantial influence on the maximum distance when the capillary bridge breaks. In the case presented here, this maximum distance for the constant volume is about 4 times higher than that for the modeling in thermodynamic equilibrium. Another important piece of information is how the capillary force is divided into capillary pressure force and surface tension force. In general the force based on the capillary pressure is considered to be dominant over the surface tension force.23 This is the case for micrometer sized particles and the respective capillary bridges. The surface tension can even be neglected without a substantial mistake. But as the particles become smaller, the surface tension force becomes more important, which can be seen in Figure 10. The capillary bridges between smaller particles are in general smaller than between bigger particles. If the cross-section of the bridge is smaller, the perimeter, and thus the surface tension force, becomes more important compared to the cross-sectional area, and thus the capillary pressure force. Furthermore, with increasing relative humidity the ratio of the capillary pressure force declines as the capillary pressure also decreases with growing relative humidity and Kelvin radius.
Figure 8. Total capillary force FC depending on separation distance d and relative humidity φ for capillary bridges between two spheres with a diameter D of (a) 100 nm and (b) 10 nm and a contact angle Θ of 0° and a surface tension of the liquid γ of 72.74 mN/m at equilibrium.
capillary force is at a maximum, and it decreases steadily while the particles are separated from each other. At a certain distance the capillary bridge ruptures. This rupture distance increases with higher relative humidity as more water condenses in the meniscus. At high relative humidities the capillary force is not as dependent on the distance as at lower humidities. Due to the higher capillary pressure, the force at direct contact is slightly higher at lower humidities as was mentioned before. When separating two particles, two different cases have to be considered: (1) The capillary bridge is during the whole process in equilibrium with the surrounding air, or (2) the volume of the fluid phase remains constant. In the case of a particle pull-off from a plate, the results of these two possibilities differ very much, as can be seen in Figure 9. First the force and the volume were calculated when particle and
4. CONCLUSION In this paper a simulation was presented that calculates numerically the exact shape as well as resulting forces of a capillary bridge in the case of capillary condensation, that means it is dependent on the relative humidity of the air. 1060
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which minimum particle size the continuum assumption for the liquid bridge is still valid. Additionally, this comparison shall reveal the behavior at small relative humidities since there has to be a minimum amount of water on the surface that it behaves like a liquid phase26 and also the curvature of the meniscus may reach values in the range of molecule sizes.27 Furthermore, the simulation will be enhanced to determine also nonrotational-symmetrical bridges and the transient behavior of the meniscus. This allows the calculation of capillary bridges between particles that are not rotational-symmetrical or that have arbitrary roughnesses. Also, the forces that do not act in normal direction can be calculated, and the dynamics of particle separation in normal and non-normal direction can be determined.
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Figure 10. Ratio of capillary pressure force FP on total force FC for different particles sizes D and relative humidities φ between two equal sized spheres with a distance d of 0.2 nm.
AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected].
Particular attention has been paid to nanoscale particles, since capillary condensation is much more important in that case. To validate this model it was shown that results obtained with this simulation agree very well with several other analytical methods to calculate the capillary force. Furthermore, two approximations that are often used were compared to this work. The first approximation is to consider the meniscus profile as a circle to simplify the calculation. This assumption shows good agreement for micrometer sized particles, but as the particles become smaller the deviation to this work increases strongly since the implied assumptions are not valid anymore at this scale. Furthermore, this work predicts the existence of capillary bridges at lower humidities compared to simulations using this approximation. The second popularly used assumption is the use of an effective radius or diameter that allows one to calculate the capillary bridge between a sphere and a plate instead of the capillary bridge between two spheres. It is shown that the results differ for small particle sizes at high relative humidities only, but the shape of the capillary bridge is different, and due to the higher filling angle numerical problems can occur. Therefore, this work gives more accurate results especially for nanoscale particles and should be preferred. It was shown that capillary bridges depend on several different parameters. As the particle size increases also the resulting force increases. But the distribution in the components changes. For micrometer sized particles the capillary pressure force is dominant, whereas surface tension force becomes increasingly important for decreasing particle sizes. A higher relative humidity leads to bigger capillary bridges, suggesting a higher capillary force, but at the same time the capillary pressure decreases. Hence there is a maximum of the resulting force with respect to relative humidity. A higher contact angle, that is, a more hydrophobic surface, decreases the capillary forces, and also a higher relative humidity is necessary for the formation of a meniscus. When the particles are separated from each other the adhesion force decreases quite slowly first until no capillary bridge can exist anymore and it breaks. A comparison between two possibilities of modeling the particle separation was shown, that is, assuming the capillary bridge in thermodynamic equilibrium and assuming a constant bridge volume, respectively. It is shown that a constant bridge volume leads to the existence of a liquid bridge up to much larger distances compared to the assumption of equilibrium. In the future a comparison to results from molecular dynamics simulations will be made in order to check down to
Author Contributions
The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thank Deutsche Forschungsgemeinschaft (DFG) for the financial funding of this project in the priority program “Partikel im Kontakt−Mikromechanik, Mikroprozessdynamik und Partikelkollektive” (SPP 1486) under grant SCHM1429/9.
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REFERENCES
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