Langmuir 2008, 24, 1459-1468
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Modeling Adhesion Forces between Deformable Bodies by FEM and Hamaker Summation† Hongben Zhou and Wolfgang Peukert* Institute of Particle Technology, Friedrich-Alexander UniVersity of Erlangen-Nuremberg, D-91058 Erlangen, BaVaria, Germany ReceiVed July 30, 2007. In Final Form: October 15, 2007 A new numerical approach is presented for predicting adhesion forces of particles at flat and rough surfaces. The new hybrid method uses the finite element method (FEM) for the determination of elastic and plastic particle deformation combined with numerical Hamaker summation. In the numerical approach, the influence of the plastic deformation can be fully included. We show how the adhesion force depends on the contact geometry and the material properties. For easy comparison with other models, the force-displacement behavior of the systems is presented. The numerical approach is supported by atomic force microscopy (AFM) measurements. The experimentally observed adhesion force hysteresis is described very well by the new approach. Although calculations in this article are focused on spherical particles, our approach can be extended to particles of arbitrary shapes.
1. Introduction How is the adhesion force between a solid particle and a surface predicted? This question has been discussed for several decades. It is especially difficult for polymer particles because polymer materials have low Young’s moduli and low yield stresses. They can deform elastically and plastically under stress. This causes an increase in the contact area between the particle and the surface and hence the increase in adhesion forces, the most prominent of which is the van der Waals force. The adhesion force between particles and surfaces is relevant to many industrial processes such as powder handling, printing, cleaning of surfaces, and cell-wall interactions. The adhesion of a particle on a substrate is a very complex process that is influenced by thermomechanical material properties, topography and surface chemistry in the contact area, contact time, and external forces. Even for large particles in the range of micrometers to millimeters, the contact is governed by processes on the lower nanoscale. In this article, we are combining a continuum with a molecular approach. Continuum mechanics determines deformation and thus the shape of the contact partners in the contact region whereas the adhesion force is determined from a summation of the molecular forces. Different models have been developed to predict the adhesion force. Hamaker1 developed the microscopic theory based on the non-retarded additivity of the interaction energy. Lifshitz2 treated the interacting bodies as continuous media and derived the adhesion force as a function of the bulk properties, such as the dielectric constants. Considering the elastic deformation of the adhesion partners, Johnson, Kendall, and Robert3 and Derjaguin, Muller, and Toporov4 derived the adhesion force as a function of the so-called work of adhesion. Both theories calculate the elastic particle deformation according to Hertz.5 On the basis of Sneddon’s penetration model6 of a rigid punch with a spherical †
Part of the Molecular and Surface Forces special issue. * Corresponding author. E-mail:
[email protected].
(1) Hamaker, H. C. Physica IV 1937, 10, 1058-1072. (2) Lifshitz, E. M. SoV. Phys. 1956, 2, 73-83. (3) Johnson, K. L.; Kendall, K.; Robert, A. D. Proc. R. Soc. London, Ser. A 1971, 324, 301-313. (4) Derjaguin, B. V.; Muller, V. M.; Toporov, Y. P. J. Colloid Interface Sci. 1975, 53, 314-326. (5) Hertz, H. Reine Angew. Math. 1882, 156-171.
profile into an elastic half-space, Maugis7 suggested that the JKR and DMT models are the limiting cases of a more general model. In the JKR model, the interaction outside the area of contact is neglected whereas in the DMT model the influence of the adhesion force on particle deformation is not taken into account. Krupp and Sperling8 and Molerus9 derived models considering the influence of the purely plastic deformation on the van der Waals force. Rumpf and co-workers10,11 extended these models with elastic-plastic, linear viscoelastic, and viscoplastic material properties. All of these models assumed that the adhesion force consists of two partssinside of the contact circle and outside of the contact circle. The adhesion force in the contact circle is calculated with the Hamaker model between two surfaces. The adhesion force outside of the contact circle is assumed to be equal to the adhesion force despite the influence of deformation. Tomas12,13 compared the contact models between deformable particles in a normal (pull-off) force versus displacement diagram for the interaction between spherical particles with elastic, plastic, viscoelastic, and viscoplastic behavior. In this article, we use the finite element method (FEM) for the prediction of particle deformation combined with numerical Hamaker summation. Other molecular forces such as hydrogen bonds are neglected in this article but could be included under the assumption of additivity of forces. In the numerical approach, the influence of the plastic deformation is fully included. We show that for polymer particles in contact with a rigid surface plastic deformation may even occur without applying any external force (i.e., the acting adhesion force leads directly to plastic deformation in the contact zone). This model can be further extended to predict the adhesion force between particles and surfaces with arbitrary material properties and arbitrary shape. (6) Sneddon, I. N. Int. J. Eng. Sci. 1965, 3, 47-57. (7) Maugis, D. J. Colloid Interface Sci. 1992, 150, 243-269. (8) Krupp, H.; Sperling, G. J. Appl. Phys. 1966, 37, 4176-4180. (9) Molerus, O. Powder Technol. 1975, 12, 215-243. (10) Rumpf, H. Pharm. Ind. 1972, 34, 270-281. (11) Rumpf, H.; Sommer, K.; Schubert, H. Chem. Ing. Tech. 1976, 48, 300307. (12) Tomas, J. In Particles on Surfaces 8: Detection, Adhesion and RemoVal; Mittal, K. L., Ed.; Brill Academic Publishers: Leiden, The Netherlands, 2003; pp 183-229. (13) Tomas, J. Chem. Eng. Sci. 2007, 62, 1997-2010.
10.1021/la7023023 CCC: $40.75 © 2008 American Chemical Society Published on Web 01/08/2008
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2. Models of Particle Adhesion
ratio of the materials, respectively. Distant points in these two
2.1. Hamaker Summation Method and Lifshitz Macroscopic Theory. Hamaker1 assumed that the molecular interaction is nonretarded and additive, so the interaction energy U between two arbitrary bodies is
spheres approach each other by a distance of δ ) xF2/(RK2) ) rC2/R.5 The JKR model3 assumed a balance between the mechanical energy UM, the elastically accumulated energy UE, and the surface energy US.
U)-
∫V ∫V 1
F1F2C 2
r6
dV1 dV2
(1)
where F is the number density of the atoms in the interacting bodies, C is the London-van der Waals constant, r is the distance between the interacting atoms, and V1 and V2 are the volumes of the interacting bodies. Between two spherical bodies in contact, this interaction energy is integrated to
U)-
A R1R2 6a R1 + R2
(2)
3
UT ) UM + UE + US
(6)
Johnson et al. described the adhesion procedure in two steps. In step 1, the surface force is neglected. The system is loaded with a normal force, and the force-displacement relationship obeys the Hertz theory until the contact radius rC is reached. In step 2, the contact radius is kept constant at rC, and the normal force is replaced by the surface force to reach the end status. The balance of mechanical energies leads to the following expression for the contact radius:
a , R1 and a , R2
R rC3 ) (F + 3πRW + x6πRWF + (3πRW)2) K
where A ) π2F1F2C is defined as the Hamaker constant, R1 and R2 are the particle radii, and a is the distance between the particles. The adhesion force between these two spheres is given as
W is the work of adhesion, 6πRWF + (3πRW)2 ) 0 is the criterion for the separation, thus the force at the moment of separation is
dU Fadh ) da
|
A R1R2 ) a)a0 6a02 R1 + R2
(3)
with a0 being the minimal contact distance. On the basis of the Hamaker approach, Go¨tzinger et al.14 could predict the mean adhesion forces between nanorough alumina spheres on a flat substrate. The macroscopic theory according to Lifshitz2 avoids the problem of the assumption of additivity of forces by treating the interacting bodies as continuous media. The Hamaker constant can be expressed in terms of the frequency-dependent dielectric constants
( )( ) ∫ (
3 1 - 3 2 - 3 + A ≈ kT 4 1 + 3 2 + 3 3h ∞ 1(iν) - 3(iν) 2(iν) - 3(iν) dν (4) 4π ν1 1(iν) + 3(iν) 2(iν) + 3(iν)
)(
)
where 1, 2, and 3 are the static dielectric constants of both bodies and the medium, (iν) represents the values of at imaginary frequencies ν, and h is Planck’s constant.15 2.2. JKR Approximation. In the case of soft polymer particles, the contribution of particle deformation to the increase in the adhesion force has to be included. Elastic particle deformation can be predicted according to the Hertz theory.5 Hertz investigated the deformation in the contact region between two smooth macroscopic elastic spheres. For two spheres of radii R1 and R2 pressed together with a normal force F, the contact radius rC is given by
rC3 )
R F K
3 Fadh ) - πRW 2
(14) Go¨tzinger, M.; Peukert, W. Powder Technol. 2003, 130, 102-109. (15) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: San Diego, CA, 1991.
(8)
2.3. DMT Approximation. The DMT approximation4 is suitable for hard adhesion partners.7 Derjaguin et al. assumed that the normal force is equal to zero at the moment of separation. Equilibrium is established between the elastic energy and surface energy. In this model, the surface energy is
US ) W(a0)πrC2 + 2π
∫r∞ W(a)r dr C
where W(a) is the interaction energy per unit area, which depends on the contact distance a and W(a0) has the same definition as the work of adhesion W in the JKR model. r is the radial coordinate, and rC is the contact radius. The first term is the surface energy within the contact area; the second term describes the interaction outside of the contact area. Derjaguin et al. calculated the surface force by differentiating the surface energy with respect to the displacement δ. Separation takes place when the deformation is totally recovered (i.e., rC ) 0 and δ ) 0). The adhesion force is
Fadh ) FS(δ ) 0) ) 2πRW(a0)
(9)
2.4. Maugis Approximation. Maugis7 compared the JKR and DMT approximations in contact radius versus normal force diagrams and suggested that JKR and DMT are the limiting cases of a more general model. In the JKR model, the interaction outside the area of contact is neglected, whereas in the DMT model, the influence of the adhesion force on particle deformation is not taken into account. Maugis introduced a characteristic material coefficient λ
(5)
where R ) R1R2/(R1 + R2) is the equivalent radius and K ) 4/(3π)((1 - ν21)/(πE1) + (1 - ν22)/(πE2))-1 is the elastic constant of the system. E and ν are Young’s modulus and the Poisson
(7)
λ)
2σ0/(K/R)2/3 (πRW)1/3
(10)
where σ0 is the theoretical cohesive stress in the interacting area. For λ f 0, the DMT limit is reached. Separation takes place when the contact radius decreases to zero. When λ f ∞, the JKR limit is reached. Separation takes place when the Griffith instability is reached. In case of polymer particles, the Young’s
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moduli are relatively small. The value of the coefficient λ lies in the range of 5-10. The adhesion force approaches the JKR limit. The Maugis model is supported by Johnson and Greenwood.16 2.5. Rumpf Approximation.11 According to Hertz theory, the pressure distribution in the contact area is (p/p0)2 ) 1 (r/rC)2, where p0 ) 1.5FN/(πrC2) is the maximal pressure in the center of the circle and rC is the radius of the contact area. When p0 is larger than the Hertzian hardness pHpl, the particle deforms plastically. The plastically deformed range of the area is
fpl ) πrpl2 )
3 2
FN + Fadh,0 pHpl p0 pHpl
( ( )) 1-
force according to r3 ) (R/K)FT, and the displacement is given by δ ) r2/R (Hertz theory). Thus, the following relationship between the total normal force and the surface force holds:
A R 2/3 A FD ) πr2 ) FT 3 6πa0 6a03 K
( )
The normal force is
F N ) FT - FD ) FT -
pHpl 2
The adhesion force is
( )
(14)
˜ T2/3 F ˜D ) F
(13a)
˜T - F ˜ T2/3 ) F ˜ D3/2 - F ˜D F ˜N ) F
(14a)
(11) and
where Fadh,0 is the adhesion force without any particle deformation and pvdW fpl represents the van der Waals force of the plastically deformed range and the surface. The van der Waals pressure pvdW can be calculated with the Hamaker model between a plate and a half space: pvdW ) A/(6πa03). 2.6. Micromechanical Approach of Tomas. Tomas13 and Tyhoniuk et al.17 included plastic deformation and friction in their approach. High normal forces lead to an irreversible increase in the contact area, thus force-displacement plots are characterized by hysteresis. Constitutive equations should contain at least the following six mechanical material parameters: particle radius, Hamaker constant, Young’s modulus, Poisson ratio, microyield strength to account for plastic deformation, and friction coefficient.
3. Analytical Approach to Elastic Deformation A general drawback of the existing models is that the exact contact geometry is not known when nonlinear (i.e., plastic or viscous) effects occur. Models applying Hertz theory (e.g., the JKR model) underestimate the adhesion force by neglecting the dispersion force outside of the contact area. For a wider range of applications (e.g., for materials with nonlinear properties), numerical methods have to be applied. To illustrate the modeling strategy, in the following text an analytical model with purely elastic deformation based on Hertz theory is presented. It is assumed that the adhesion partners are brought into contact very slowly so that at every moment the energetic equilibrium between the three energy terms as described in the JKR model (section 2.2) is maintained. The deformation of the soft particle depends on the sum of the normal forces FN and dispersion forces FD. For convenience, this force is defined as the total normal force: FT ) FN + FD. In the analytical model, the relationship between the total normal force and the particle deformation is described by the Hertz model. The dispersion force FD is the force needed to separate the adhesion partners abruptly without changing the deformation of the particle. It can be calculated using the Hamaker summation method:
A FD ) πr2 6πa03
A R 2/3 FT 6a03 K
Equations 13 and 14 can be normalized by dividing the forces FD, FT, and FN by (A/6a03)3(R/K)2. We get
p0
Fadh ) Fadh,0 + pvdW fpl
(13)
(12)
Similar to the JKR model, the interaction outside of the contact area is neglected. The contact radius depends on the total normal (16) Johnson, K. L.; Greenwood, J. A. J. Colloid Interface Sci. 1997, 192, 326-333. (17) Tyhoniuk, R.; Tomas, J.; Luding, S.; Kappl, M.; Heim, L.; Butt, H.-J. Chem. Eng. Sci. 2007, 62, 2843-2864.
where F ˜ D, F ˜ T, and F ˜ N are the normalized forces, respectively. The approaching and retracing procedures can be described as follows: (a) As soon as the particle is in contact with the surface, the particle deforms until the equilibrium between the dispersion force and the elastic force is reached. (b) As the normal force increases, a new equilibrium can be established for each value of the normal force, and the dispersion force increases with the normal force. (c) The normal force is reduced, turns into a pulling force, and reaches a calculational maximum, which is defined as the critical pulling force Fpull,crit. Equilibrium can also be reached at each point on the reference curve. (d) When Fpull,crit is exceeded, equilibrium can no longer be established. Adhesion becomes instable, and the surfaces separate abruptly, even though the dispersion force FD is still larger than the pulling force. Griffith18 observed this energy instability. Cracks occur if they happen to reduce (or maintain) the sum of the potential energy of the applied force and the strain energy of the body. The application of Griffith theory in predicting the separation of adhesion partners is supported by Tabor19 and Maugis.7 For the separation, it is necessary to overcome the critical pulling force. This force can be reached if
dF ˜N 3 1/2 )0w F -1)0 ˜ dF ˜D 2 D
(15)
By solving eq 15, we get the normalized dispersion force and the normalized normal force F ˜ D ) 4/9 and F ˜ N ) -4/27, respectively. The critical pulling force is then
( )( ) ( )( )
Fpull,crit ) -F ˜N )
A 6a03
4 A 27 6a 3 0
3
3
R K
R K
2
2
(16)
Fpull,crit is the adhesion force measured with different methods (e.g., force-distance measurement with AFM or the centrifugal detachment method). For comparison, the curves according to (18) Griffith, A. A. Philos. Trans. R. Soc. London, Ser. A 1921, 221, 163-198. (19) Tabor, D. J. Colloid Interface Sci. 1977, 58, 2-13.
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Figure 1. Normalized force equilibrium plots of the analytical model in comparison to the JKR model with different Young’s moduli.
the JKR model are also shown in Figure 1. The dispersion force is calculated with eq 12. The contact radius rC is a function of the normal force FN according to eq 7. The Hamaker constant A is related to the work of adhesion W by A ) 12πa02W according to Israelachvili,15 where a0 ) 0.165 nm is the equilibrium separation of the atoms. Here we choose a value of the work of adhesion W ) 0.064 J/m2; accordingly, the Hamaker constant is A ) 6.6 × 10-20 J. The position of the JKR curve in the normalized F ˜D - F ˜ N equilibrium plot is not unique. It depends on the values of the single parameters (e.g., the Young’s modulus) as shown in Figure 1.
4. Numerical Model for the Prediction of the Adhesion Force 4.1. FEM Simulation of the Particle Deformation. In this study, FEM simulations of the particle deformation were carried out with the program Abaqus 6.4-1. The particle was simulated with an axisymmetric four-node model. In some cases (especially in the case of particle-asperity contacts), grid sizes of below 1 nm in the contact region are necessary for sufficient accuracy. This fine grid size nevertheless leads to meaningful results, although on this small scale the application of any continuum model is questionable in the strict sense. From molecular dynamics simulations of the compression of a spherical crystal, one may conclude that the lower limit is around 3-5 nm.20 Luan et al.21 even state that continuum mechanics may be applicable down to lengths as small as two or three atomic diameters. However, the atomic structure of surfaces can have profound consequences for larger contacts. The (macroscopic) mechanical material properties of the particle are described in this article by elasticity or bilinear isotropic plasticity (i.e., linearly elastic under the yield stress and then linearly plastic with a reduced Young’s modulus; see Figure 4). The total normal force FT is simplified to a force acting on a distant point on the rotation axis. Among the adhesion partners considered in this study, the flat or rough substrate surfaces always have significantly higher Young’s moduli than the particles, hence they are considered to be rigid in this model. In the FEM simulation, the stress in all directions in space can be normalized to an overall stress, the von-Mises stress: (20) Miesbauer, O.; Go¨tzinger, M.; Peukert, W. Nanotechnology 2003, 14, 371-376. (21) Luan, B.; Robbins, M. O. Phys. ReV. E 2006, 74, 026111.
Figure 2. Dispersion interaction energy calculated in two steps: (1) integration according to Hamaker1 to get the point-to-sphere interaction energy (top) and (2) numerical integration over the deformed particle (dotted area) to get the total interaction energy (bottom).
σ)
x
σxx2 + σyy2 + σzz2 (σxxσyy + σyyσzz + σzzσxx) + 3(σxy2 + σyz2 + σzx2)
(17)
The von-Mises stress is equivalent to the absolute value of the tensile stress in the tensile test. The local deformation of the material depends on the local von-Mises stress. The total normal force FT is varied in the FEM simulation. If the total normal forces are applied, then the coordinate of each node on the particle surface can be exported to data files, which constitute the input file for the calculation of the dispersion force. 4.2. Calculation of the Dispersion Force. As derived from the Hamaker model (eq 1), the dispersion interaction energy between two interacting axisymmetric bodies is
U)-
∫z ∫z ∫r ∫r ∫θ ∫θ 2
1
2
1
2
CF1F2r1r2 1
S6
dθ1 dθ2 dr1 dr2 dz1 dz2 (18)
Here, F1 and F2 denote the number densities of the molecules in both bodies and C and S are the London-van der Waals constant and the distance between unit volumes, respectively. The unit volume is expressed in a cylindrical coordinate system as dV ) r dθ dr dz.
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Figure 3. Numerically simulated force equilibrium plot of a 10 µm polystyrene particle in contact with a flat, rigid surface with the assumption that the particle deformation is elastic. For comparison, the results of the analytical approach (eqs 13 and 14) are also illustrated. Separation occurs when the pulling force reaches Fpull,crit.
The interaction energy between a point P in the deformable particle and the rigid sphere (Figure 2, top) can be calculated by integrating the interaction energy over a rigid sphere of radius R1:1
Upoint_sphere ) -
∫SS+R -R
1
(
1
CF r 2 π [R - (S - r)2] dr 6 S 1 r
2 2 CFπ 2S R1 - S 1 ) + - 2 4 S 3r3 4r 2r
)|
r)S + R1 r ) S-R1
Further integration over the deformable particle has to be carried out numerically (e.g., by applying the rectangle rule):
Uparticle_sphere ) 2πF
) 2A
[(
× ∫x ∫z CFπ S
[(
1 2S
∑x ∑z S
3r3
)| ]
2 2 1 2S R1 - S + - 2 3 4 3r 4r 2r
+
R12 - S2 4r4
-
r)S + R1
x dx dz
r)S-R1
)| ]
1 2r2
r)S + R1
x ∆x ∆z
r)S-R1
The distance S from a point in the deformable particle to the center of the rigid sphere can be calculated from the x and z coordinates of this point (Figure 2, bottom). In the case of a micrometer-sized particle in contact with a rigid sphere or surface, x ) 0...200 nm and z ) zmin(x)...100 nm is sufficient to keep the relative error below 1%. zmin(x) describes the contour of the deformed particle, which is achieved by interpolating the discrete contact geometry data from the FEM simulation with a cubic spline, taking into account a contact distance a0 ) 0.4 nm. Of course, dispersion forces inside and outside of the region of direct contact (at a0) are included. The dispersion force is obtained from the derivation of the dispersion interaction energy with respect to the contact distance. Different total normal forces FT are applied to the particle to obtain a curve in the force equilibrium plot (Figure 3). It is important to note that this approach is not limited to spherical geometries but can be applied to any contact geometry. Intrinsically, this hydrid approach combines continuum modeling of material deformation with molecular simulation to obtain the adhesion forces. Under the assumption of additivity of dispersion
Figure 4. Stress-strain diagram of polystyrene 158K (BASF) at 23 °C and the approximation of this curve with the bilinear model.
forces, the molecular simulation is replaced by a volume integral over the contact geometry. Generalizing this approach to other thermomechanical properties, different particle sizes and shapes, other types of intermolecular forces, and varied external forces is straightforward. 4.3. Case Study 1: Adhesion of a Polystyrene Particle on a Flat Rigid Surface. The adhesion between a polystyrene particle with a low Young’s modulus and a flat rigid surface (R f ∞ and E f ∞) is simulated according to the numerical approach. At first, it is assumed that the deformation of the polystyrene particle is purely elastic. The data used for the simulation of the particle deformation and for the calculation of the adhesion force are as follows: particle diameter, 2R ) 10 µm; Young’s modulus, E ) 3.3 GPa; Poisson number, ν ) 0.32; and Hamaker constant, A ) 6.6 × 10-20 J. The Young’s modulus and the Poisson number are taken from the CAMPUS database for polystyrene 158K (BASF) at 23 °C. The value of the Hamaker constant is taken from Visser22 applying the Lifshitz approach. The FEM method has been validated by comparison with Hertzian contact mechanics. Both approaches should lead to (22) Visser, J. In Surface and Colloid Science Matijevic, E., Ed.; Wiley: New York, 1976; pp 3-84.
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Figure 5. Numerically simulated force equilibrium plot of a 10 µm polystyrene particle in contact with a flat, rigid surface, where the material property of the particle is modeled with bilinear plasticity.
similar results as long as the defomation of the sphere remains purely elastic. This has been verified for the given spherical polysterene bead on a flat surface being compressed by a total normal force of FT ) 5 µN. Both approaches lead to an identical axial displacement of 6 nm and to contact radii of 180 and 172 nm for the FEM and Hertzian methods, respectively. The numerically obtained force equilibrium curves are shown in Figure 3 together with the corresponding analytical solutions resulting from eqs 13 and 14. It can be observed that the dispersion forces calculated with the numerical method are significantly higher than the analytical results. Accordingly, the adhesion force (i.e., the critical pulling force) is also higher. This difference is due to the restrictions of the analytical model, which neglects the dispersion force outside of the contact area. Obviously, this difference is smaller if the Young’s modulus E is smaller (i.e., the contact radius is larger). In the case of a 10 µm polystyrene particle, the FEM simulation shows that the yield stress of 50 MPa is reached with a total normal force of 5 µN. Without any external normal force (i.e., FN ) 0), the dispersion force reaches 8.3 µN. The polystyrene particle deforms plastically as soon as contact to a flat, rigid surface is established. Therefore, it is necessary to take the plastic deformation of the particle into consideration. The plasticity of the material is simulated with a bilinear plasticity model. In the case of polystyrene, the Young’s modulus is 3.3 GPa in the elastic range. If the stress is larger than the yield stress of 50 MPa, then the slope of the stress-strain curve is 1.0 GPa. These values are obtained through discretization of the stress-strain curve of polystyrene 158K (BASF) at 23 °C (CAMPUS database), as shown by the dotted and the dashed curves in Figure 4. As an example, the compression of a 10 µm polystyrene particle is pressed to a flat, rigid surface with a maximal total normal force of FT ) 24 µN, and the subsequent retrace procedure is simulated. As shown in the force plot (Figure 5) as well as in the force-displacement diagram (Figure 6), the approaching and retracing curves deviate from each other because the deformation cannot be totally recovered. Displacement is defined similarly to the displacement in the JKR model (section 2.2): it is the distance by which distant points in both adhesion partners approach each other. The normal force in the force-displacement diagram (Figure 6) becomes negative at small displacements. This effect can be explained by the simulation routine. During the FEM simulation,
Figure 6. Numerically simulated force-displacement behavior of a 10 µm polystyrene particle in contact with a flat, rigid surface.
a total normal force of increasing magnitude is applied to the particle, which causes the deformation of the particle. The dispersion force between the deformed particle and the substrate is then calculated by means of Hamaker summation. If the value of the dispersion force is higher than the total normal force applied in the FEM simulation, then the normal force becomes negative according to eq 14. The deformation at that total normal force is not stable, and the particle will deform further because of the dispersion force until equilibrium is reached (i.e., FD ) FT and FN ) 0). In the simulation including plastic deformation, the adhesion force is larger than in the elastic case (Figure 3). The value of the adhesion force depends on the maximally applied normal force. In comparison to the Rumpf model for plastic deformation (section 2.5), the numerical approach is physically more accurate because it includes the full plastic deformation and calculates the van der Waals force inside and outside of the contact region. Furthermore, it can be expanded to include arbitrary material properties and geometries. 4.4. Case Study 2: Adhesion of a Silica Particle on a Flat, Rigid Surface. A second numerical simulation is carried out with a silica particle and a flat wafer. The deformation of the silica particle is ideally elastic. We used the following data for the simulation: particle diameter, 2R ) 10 µm; Young’s modulus, E ) 75 GPa;
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Figure 7. Numerically simulated force equilibrium plot of a 10 µm silica particle in contact with a flat rigid surface in comparison to the results of the analytical approach (eqs 13 and 14).
Poisson number, ν ) 0.17; and Hamaker constant, A ) 6.6 × 10-20 J. The mechanical properties of silica are taken from ref 23, and the Hamaker constant is taken from ref 15. The simulation results are shown as force equilibrium curves in Figure 7. Figure 8 shows the related force-displacement diagram. It can be observed in this diagram that the separation takes place when the total normal force is reduced to zero, whereas in the case of soft polystyrene particles, as shown in Figure 6, the separation occurs before the total normal force vanishes. In this simulation, the approaching and retracing curves are identical because of the fully reversible elastic deformation. During retrace, the dispersion force reduces monotonically with the normal force (Figure 7b) until the sum of these two forces reaches FT ) 0 (Figure 7a). No critical pulling force Fpull,crit can be observed in this case. The particle separates abruptly from the surface when the pulling force Fpull surpasses the dispersion force FD. The simulation indicates that the separation does not take place before the contact radius r and the displacement δ are reduced to zero. A small deformation with r ) 8 nm and δ ) 0.36 nm remains at the moment of separation. Obviously, the continuum FEM simulation is (in the strict sense) no longer valid on such a small scale. Nevertheless, the corresponding simulation gives a reasonable adhesion force value of 354 nN,
which is only slightly larger than the result obtained from the Hamaker model (344 nN) for rigid adhesion partners. Recently, Yang et al.24 showed the good agreement of their molecular dynamics simulation with the Hertz theory on the submicrometer scale. The pressure distribution in the contact region between a spherical tip and a flat surface estimated by both approaches is quite similar. The continuum approaches (e.g., the Hertz model) do not completely lose their validity, except that the continuum models always assume a sudden jump of the contact pressure to zero at the edge of the contact circle whereas the molecular dynamics simulations predict a physically more reasonable, gradual reduction of the contact pressure along the radius. Independent of the work of Yang et al., Luan and Robbins21 simulated the pressure between a tip and a flat surface with molecular dynamics simulation. The local contact pressure is very sensitive to the molecular structure of the surface. In the case of an amorphous structure, for instance, the local contact pressures and the substrate scatter strongly. However, average contact pressures agree fairly well with Hertz theory. Both Luan and Robbins and Yang et al. found that the radius of the contact region is slightly larger than the prediction of the Hertz model. As shown in section 4.3, the FEM simulation also predicts a slightly larger contact radius than does the Hertz model. The case studies show that separation takes place if one of the two criteria is fulfilled. Between smooth adhesion partners, the type of separation first depends on the material properties of the adhesion partners. Adhesion partners with low moduli are separated if the critical pulling force Fpull,crit is reached (type A). Adhesion partners with high moduli can be separated if Fpull ) FD (type B) (i.e., before the critical pulling force Fpull,crit is reached). 4.5. Case Study 3: Adhesion of Polystyrene Particle on Rough Surfaces. Small asperities in the contact region have a significant influence on the adhesion force between particles and substrates. Rumpf10 showed that for rigid adhesion partners the adhesion force is reduced by a factor of 100 in the presence of an asperity of the proper size, which is usually in the nanometer range. Because of the small contact area, the stress is concentrated in the contact region. The maximal von-Mises stress in the contact region is significantly higher than in the case of contact between
(23) Kuchling, H. Taschenbuch der Physik; Fachbuchverlag: Leipzig, Germany, 1999.
(24) Yang, C.; Tartaglino, U.; Persson, B. N. J. Eur. Phys. J. E 2006, 19, 47-58.
Figure 8. Numerically simulated force-displacement behavior of a 10 µm silica particle in contact with a flat rigid surface.
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Figures 10b and 11b. In this case, the adhesion force (0.43 µN) is significantly smaller than for the case with higher normal forces. In conclusion, it is possible to adjust the adhesion force and the type of separation (i.e., A and B, respectively) by varying the maximal normal force and the size of the asperity. 5. Experiments
Figure 9. Maximal von-Mises stress in the contact region of a polymer particle in contact with a nanosized asperity. The literature values of the Young’s modulus and yield stress of different polymer materials are shown as circles in the diagram.
smooth adhesion partners. Figure 9 shows the maximal vonMises stress as a function of the Young’s modulus of the particle and the total normal force for the case of a 10 µm particle in contact with an asperity of r ) 250 nm. The material properties of different polymer materials (Young’s modulus, yield stress) (chosen from the CAMPUS database) are also shown in this Figure. Obviously, the yield stress of the polymers can be reached with a fairly small total normal force FT, which is below the adhesion force between the particle and the substrate. Furthermore, the maximal von-Mises stress is approximately proportional to the inverse quadric asperity radius as obtained from simulations (not shown in this article) where the asperity size was varied systematically. With decreasing asperity size, yielding occurs at even smaller total normal forces. Thus, the influence of the plastic deformation on the particle adhesion between rough adhesion partners is a major influencing parameter. In the next step, the adhesion force between a 10 µm polystyrene particle and a rigid surface with an asperity of r ) 50 nm in the contact region is simulated. As in the simulation between smooth adhesion partners, the material property of polystyrene is modeled with bilinear plasticity. The simulation with a maximal normal force of FN ) 3.2 µN is shown in Figure 10a. In comparison to the simulation between smooth adhesion partners, there is a range of small normal force (a f b) where the asperity acts as a spacer between the particle and the flat surface (i.e., the particle still has no direct contact with the flat surface or the contact is so weakselastic deformation prevailssthat it will automatically break if the normal force disappears). This case is shown in Figure 11b; the sudden increase in the slope at the end of the curves indicates that the particle is in contact with the flat surface. Figure 12 shows the direct contact between the polystyrene particle and the flat surface completely covering the asperity in the contact region. If the normal force exceeds point b, then the particle deforms so strongly that the asperity losses its function as a spacer. Similar to step a described in section 3, a new equilibrium will be reached at point c with a significantly higher dispersion force. The next steps (c f d f e) of the procedure are the same as described in section 3. The separation is of type A, where a maximum of the pulling force (or a minimum of the normal force) can be observed in Figure 10a as well as in Figure 11a. The adhesion force is equal to the critical pulling force at point e (1.2 µN). The case of normal forces in the region from a to b is shown in Figure 10b. A separation of type B takes place at point e′, when Fpull ) FD; accordingly, FT ) 0, as shown in
Adhesion forces between particles and substrates can be experimentally determined by means of atomic force microscopy (AFM) to high precision.25 The first force measurements with AFM were reported at the end of the 1980s.26,27 Cappella25 reviewed the measurement of the force-displacement curve in detail. His review covers the theoretical background and the experimental methods of this measuring technique, including the determination of the spring constant of the cantilever and mounting the particle on the tip of the cantilever. They also give a brief description of the forces involved. Butt et al.28,29 reviewed the recent efforts in understanding the adhesion phenomena by means of AFM measurements, including particle-particle interaction in a liquid. In this study, AFM measurements were carried out between smooth, surface-untreated polystyrene particles (Postnova, Landsberg) with smooth (inspection by SEM) 10-µm-diameter and flat silicon wafers (Wacker, Burghausen) with a commercial AFM (Nanoscope IIIa, Veeco) equipped with a liquid cell. Before the measurement, the particle is rinsed with ultrapure water and dried with pure nitrogen. The silicon wafer is also rinsed and dried, and afterward it is heated to 800 °C to remove the adsorbate layers on the surface and then cooled to room temperature in a pure nitrogen atmosphere. A tipless cantilever of the type CSC12 (Mikromasch, Tallinn, Estonia) is used in this measurement. The spring constants of the cantilever are determined by the added mass method.30 To exclude the influences of ambient humidity, the liquid cell is flushed with pure nitrogen (Messer Griesheim, 5.0) for at least 10 min before the measurement as well as during the measurement. Force-distance curves were measured with 1.2 µm/s approaching and withdrawal velocity. To let the particle deformation reach its equilibrium state, the reverse delay time was set to 5 s. It can be observed in the AFM measurements that the contact time has a significant influence on the adhesion force, when the contact time increases from several hundred milliseconds to approximately 1 s. However, upon further increase of the contact time (e.g., from 3 to 20 s, the variation of the adhesion force becomes minor, with relative errors below 3%. Similar tendencies are also observed by the group of Buzio et al.31 in their AFM measurements with elastomer poly(dimethylsiloxane) particles. However, poly(dimethylsiloxane) is a very soft and viscous material because of its very low glass-transition temperature of Tg ≈ -125 °C.32 The time needed to reach equilibrium (approximately 100 s) is significantly higher than that of polystyrene. Because of the long contact time and the low withdrawal velocity, particle deformation can be considered to be purely elastic-plastic, and the time dependency of the particle deformation has been neglected. Starting with a value of 2.5 µN, the normal force was gradually increased to 9 µN and finally reduced to 5 µN. At each normal force, the adhesion force was measured at least 30 times. The measured results with standard deviations (usually below 2.5%) are shown in Figure 13. In comparison, the simulation results of two particles with different values of the Young’s modulus are shown as dashed curves. In both cases, plastic deformation is included (25) Cappella, B.; Dietler, G. Surf. Sci. Rep. 1999, 34, 5-104. (26) Meyer, E.; Heinzelmann, H.; Gru¨tter, P.; Jung, T.; Hidber, H. R.; Rudin, H.; Gu¨ntherodt, H. J. Thin Solid Films 1989, 181, 527-544. (27) Weisenhorn, A. L.; Hansma, P. K.; Albrecht, T. R.; Quate, C. F. Appl. Phys. Lett. 1989, 54, 2651-2653. (28) Butt, H. J.; Cappella, B.; Kappl, M. Surf. Sci. Rep. 2005, 59, 1-152. (29) Kappl, M.; Butt, H. J. Part. Part. Syst. Charact. 2002, 19, 129-143. (30) Cleveland, J. P.; Manne, S.; Bocek, D.; Hansma, P. K. ReV. Sci. Instrum. 1993, 64, 403-405. (31) Buzio, R.; Bosca, A.; Krol, S.; Marchetto, D.; Valeri, S.; Valbusa, U. Langmuir 2007, 23, 9293-9302. (32) Lo¨tters, J. C.; Olthuis, W.; Veltink, P. H.; Bergveld, P. J. Micromech. Microeng. 1997, 7, 145-147. (33) Cleaver, J. A. S.; Looi, L. Powder Technol. 2007, 174, 34-37.
Adhesion Forces between Deformable Bodies
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Figure 10. Force equilibrium plot of a 10 µm polystyrene particle in contact with a rigid surface with an asperity of r ) 50 nm, where the material property of the particle is modeled with bilinear plasticity: (a) at large normal force and (b) at small normal force.
Figure 11. Numerically simulated force-displacement behavior of a 10 µm silica particle in contact with an asperity of r ) 50 nm: (a) at large normal force and (b) at small normal force.
Figure 12. FEM simulation. Particle deformation and local stress in a 10 µm polystyrene particle in contact with a rigid surface with an asperity of r ) 50 nm on the contact area. The particle is in direct contact with the flat surface at a total normal force of 2000 nN. when the stress is larger than 50 MPa, and the slope of the stressstrain curve is reduced to 1.0 GPa in the plastic regime, as shown in Figure 4. Also, the result according to the JKR model is shown in this diagram. Because of the low Young’s modulus of poly-
Figure 13. Adhesion force between a polystyrene particle and silicon wafer as a function of the applied normal force. AFM measurement in comparison with two series of simulations with different Young’s moduli as well as with the results according to the JKR model. (dimethylsiloxane), Buzio et al.31 measured very high adhesion forces. Unfortunately, the dependency of the adhesion force on the applied normal force is not investigated. Cleaver and Looi33 measured the adhesion force between a polystyrene particle and a polystyrene substrate with AFM. They also observed in their measurement a
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Figure 14. Prediction of the adhesion force between a 10 µm polystyrene particle and a rough substrate as a function of the asperity radius according to the FEM simulation and the Rumpf model. significant increase in the adhesion force along the applied normal force and the onset of plastic deformation. Because the adhesion partners are both smooth and homogeneous, the measured adhesion force distributions were very narrow (error bars in Figure 13). The measured adhesion force hysteresis is fully consistent with the simulation results. Furthermore, the values of the adhesion forces are also very close to the simulation results, and the calculated adhesion forces deviate less than 10% from the measured values. The described methodology can even be used as a recipe for a model-based determination of the often unknown particle properties. The JKR model (section 2.2) shows no dependence of the adhesion force on the normal force, and the model predicts smaller adhesion forces than does the simulation. The Rumpf model (section 2.5) predicts adhesion forces that are much too low, below 500 nN, because it neglects elastic deformation and strongly underestimates plastic deformation. It is well known that adhesion forces and powder flow can be controlled by nanoscale asperities. The Rumpf model for a spherical particle with a hemispherical asperity in contact with a smooth wall10 assumes that the asperity reduces the adhesion force of the particle considerably. In this case, the adhesion force is given by FvdW )
(
)
A r R + 6 (a + r)2 a 2 0 0
where A is the Hamaker constant, R and r are the radii of the particle and the asperity, respectively, and a0 is the contact distance. Any deformation is disregarded. Is this effect also valid for the case of soft particles with plastic deformation? Figure 14 shows that essentially similar behavior can be observed for hard and soft particles (i.e., the adhesion force as a function of the asperity radius goes through a distinct minimum that is located near 20 nm). Although the location of the minimum (asperity size) remains essentially unchanged, the reduction of the
adhesion force is less pronounced for soft spheres in comparison to that for hard spheres. These principal findings can be applied to more complex situations (i.e., a surface covered with nanoparticles). On rough surfaces, the adhesion forces of hard oxide particles are widely distributed,34 depending on the local contact geometry. Thus, adhesion force distributions have to be considered. Measured adhesion force distributions of a 10 µm polystyrene particle and silicon wafers coated with 34, 110, and 250 nm SiO2 nanoparticles by dip coating35 show trends similar to those observed for hard spheres. In all cases, Weibull distributions were observed as expected from theory. The minimal value of the adhesion force is approximately the adhesion force between the particle and a single asperity, which can be estimated by means of numerical simulation. Adhesion force distributions are provided as Supporting Information. The slope of the distribution depends on the structure of the coating. The adhesion force distribution was calculated in ref 34 for an ideal cubic arrangement. A similar approach could be applied for soft spheres, including plastic deformation.
6. Conclusions Particle-substrate adhesion forces can be predicted by a hybrid method combining FEM simulation and numerical Hamaker summation. This method can be applied to smooth as well as rough adhesion partners of arbitrary shape with different material properties. This method provides significantly higher accuracy than do the analytical models because the complete contact geometry is taken into account. The AFM measurements between a smooth polystyrene particle and a silicon wafer show adhesion force hysteresis that is in good agreement with the results of the simulation. We show that the adhesion force between nonrigid adhesion partners depends significantly on the normal force and on the mechanical property of the material (e.g., the elasticity and the plasticity). For easy comparison with other models, the force-displacement behavior of the systems is presented. Roughness on the nanoscale leads for most polymers to plastic deformation (i.e., the stress in the contact zone exceeds the yield stress under the influence of the adhesion force. The reduction of adhesion forces of soft particles by roughness follows similar trends as for hard particles, although the effective minimum is less pronounced. Acknowledgment. The financial support of Bayerische Forschungsstiftung and Oce´ Printing Systems GmbH is gratefully acknowledged. Supporting Information Available: Adhesion force between rough surfaces. This material is available free of charge via the Internet at http://pubs.acs.org. LA7023023 (34) Go¨tzinger, M.; Peukert, W. Langmuir 2004, 20, 5298-5303. (35) Gu¨nther, L.; Peukert, W. Colloids Surf., A 2003, 225, 49-61.
