Van der Waals Force Computation of Freely Oriented Rough Surfaces for Micromanipulation Purposes

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1 The 00 IEEE/RSJ International Conference on Intelligent Robots an Systems October 8-, 00, Taipei, Taiwan Van er Waals Force Computation of Freely Oriente Rough Surfaces for Micromanipulation Purposes Mariaana Savia, Member, IEEE, Quan Zhou, Member, IEEE Abstract This paper escribes a numerical metho to compute van er Waals force between comple an freely oriente objects that have rough surfaces. The force calculation is implemente using the surface formulation of the force. First, the interacting surfaces are approimate with triangular meshes. Then, the surface roughness is taken into consieration by separately aing the asperity heights to each point of the smooth reference surfaces. Three ifferent moels are use to escribe the rough surfaces: sinusoial, fractal, an probabilistic. All the three moels show consierable reuction in the force values as a function of the roughness peak height or the inclination angle of the surfaces. Despite the reuction of the force values for rough surfaces, van er Waals force is consiere to be an essential part when calculating the total interaction force for micromanipulation purposes. A I. INTRODUCTION LL ENGINEERING surfaces are rough at some level. At microscale, even small surface asperities influence the forces that govern the object interaction. Therefore, it is essential to take roughness into consieration in the microsystem esign. In this paper, the intene application area of the roughness calculations is contact micromanipulation. The term micro refers to object sizes below hunres of micrometers an contact micromanipulation means that the tool physically touches the objects uring manipulation. Due to the so calle scaling effect [, ], the three forces that ominate the interaction at microscale are van er Waals force, capillary force, an electrostatic force [3]. These surface forces make micromanipulation iffer consierably from manipulation at the conventional macroscale. Each one of the three forces is affecte by the surface roughness. Van er Waals force in particular is sensitive to surface roughness since the force values epen strongly on the istance between the interacting objects. In this paper, the emphasis will be entirely on van er Waals force. In terms of interaction force moels, the manipulationspecific requirement is the applicability of the moels to comple object shapes an arbitrary object orientations. This kin of moels coul be implemente for manipulation M. Savia is with the Tampere University of Technology, Department of Automation Science an Engineering, P.O. Bo 69, 330 Tampere, Finlan (phone: +358 ( ; fa: +358 ( ; mariaana.savia@tut.fi. Q. Zhou, is with the Aalto University, School of Science an Technology, Department of Automation an Systems Technology, P.O. Bo 5500, Aalto, Finlan. ( quan.zhou@tkk.fi. esign (e.g., esign of manipulation strategies, or eneffectors, for simulator evelopment, an sometimes even for manipulation control. Many types of van er Waals force moels for perfectly smooth surfaces have been presente in the literature. Propose approaches inclue both analytical (e.g., [3, 4] an numerical moels (e.g., [5, 6]. However, only a few of the publishe moels (e.g., [7, 8] satisfy the manipulationrelate requirements concerning the freeom of shape an orientation of interacting objects. An analytical metho to inclue the effects of roughness in the van er Waals force evaluation was presente in [4]. The metho takes into consieration the height of the highest roughness peak but not the ensity of the peaks. Alternative approaches utilize elementary shapes to moel the asperities, e.g., spheres an cones [9, 0]. In aition, sinusoial functions have been consiere []. Other interesting approaches to surface roughness moeling at micro- an nanoscales can be foun in tribology. In tribology, the roughness moels are categorize to probabilistic moels (e.g., Gaussian [] an fractal moels [3, 4]. In these approaches, roughness moels are typically evelope for two nominally flat parallel surfaces. As shortly iscusse above, literature shows some methos to compute van er Waals force for comple object shapes having arbitrary orientations an other methos to inclue surface roughness in the force values. However, it seems that no metho publishe this far iscusses how to combine these two important features for van er Waals force. A moel that takes into account both surface roughness an freeom of orientation woul be very valuable for micromanipulation purposes. This paper proposes a numerical metho to compute van er Waals force for freely oriente rough surfaces. The rest of the paper is organize as follows. First, Section II presents the parameterization that allows calculations for arbitrary shapes an orientations for smooth surfaces. The ifferent moels to present rough surfaces an their inclusion to the force moel are presente in Section III. Section IV gives force results an Section V iscusses the issues relate to moeling. Section VI conclues the paper. II. FORCE CALCULATION BETWEEN SMOOTH SURFACES In case of pairwise aitivity an nonretare forces, the van er Waals force between two objects can be calculate using the so calle Hamaker s approach [5, 6]. This leas to the volume formulation of the force, i.e., the total force F is /0/$ IEEE 56

2 F ρ ρ wv V, ( = V V where ρ i an V i refer to the number ensity an volume of the interacting object i =,, an w stans for the van er Waals force between a pair of atoms or small molecules, one taken from each of the interacting volumes. In our work, we have aopte the surface formulation [6] for the van er Waals force. When compare to the conventional volume formulation, the surface formulation reuces the 6-imensional integral in ( to a four imensional one. This change significantly reuces the computational compleity that is involve in the force evaluation. Surface formulation gives the total force F between two interacting objects as follows [6] F = ρ ρ ( G n n S S, ( S S where S i an n i represent the bounary surface an outwar pointing unit normal of object i =,, respectively. The function G epens on the relative orientation of the objects an on their istance from each other, i.e., C G = 3( 3, (3 where C is a material epenent constant an is a vector from point l on surface S to point k on surface S (see Fig. 3a. The integration in ( goes over all points on each surface. In our previous work, we applie the surface formulation in a parametric omain, which enables the force evaluation between comple object shapes an arbitrary orientations [7, 7]. Parameterization of the surfaces is implemente by approimating the surfaces with triangular meshes. For a pair of interacting triangles, the force in ( takes the following form F ( v ( v T = ρρ ( E E G ( E u v u v, E where E ij means ege j =, on triangle i =, an the subscript T in F T refers to the van er Waals force between two triangles, one from each of the interacting surfaces. The total van er Waals force F is then obtaine by summing z E V 0 E a b v = 0 u = 0 (-v Fig.. A triangle taken from the surface approimation of object a using Cartesian coorinates an b the parameterize triangle. v small sies of a right triangle u (4 together all the intermeiate results from ifferent triangle pairs. After the parameterization, the vector in (3 becomes =, (5 l k = V0 + ue + ve V0 ue ve where V 0i refers to a verte point between eges E ij on triangle i an u i an v i are variables in the parametric omain for triangle i (see Fig.. In practice, the force evaluation using (4 requires numerical computation (applies to ( an ( as well. We have use a lattice metho to solve the numerical quarature [8]. In a few special cases, an analytical solution to the van er Waals force in ( can be foun. An eample of a close form solution is the force for a unit area between two infinite (parallel planes at istance from each other F H =, (6 6 π 3 where H is the material epenent Hamaker constant that epens on C an on the number ensities ρ i in the following way H = π Cρ. (7 ρ The few eisting analytical solutions are useful since they can be utilize for valiating the numerical results. When comparing the force values from (4 to analytical solutions, the numerical metho gives satisfactory results, especially for flat surfaces (see also Fig. 4, Fig. 5, an Fig. 8. For curve surfaces, the metho suffers from inaccuracies that are cause by the inability of triangles to approimate curve surfaces [7, 7]. This problem can be reuce by increasing the number of surface-approimating-triangles. III. MODELS FOR ROUGH SURFACES In this paper, we consier two types of rough surfaces: eterministic surfaces that are homogeneous an isotropic an ranom surfaces. For ranom surfaces, two ifferent approaches are chosen: a fractal surface presentation where the force calculation irectly utilizes the topography information in orer to form vector in (5 an a probabilistic surface presentation where the peak height is regare as a ranom variable having Gaussian istribution an the asperity height at each point of the surface is given by a probability ensity function. Deterministic surfaces are generate using sinusoial functions so that the peak height z at each point (, y on the surface is given as [ sin ( f + cos ( y f ] a, z(, y = π π (8 where frequency f i gives the number of signal perios on ege i an a is the amplitue of the peaks. When consiering engineering surfaces, this type of a surface topography may seem unrealistic. However, it allows a straightforwar way to stuy the effects of peak height an peak ensity on van er Waals force. Fractal surfaces are generate by using the Weierstrass- Manelbrot (W-M fractal function [9]. A three- 563

3 imensional surface can be generate by using a moifie two-variable W-M function. It gives the peak height z at each point (, y on the surface as follows [0] G z(, y = L( L D n πγ ( + y cos L ln γ ( M M nma ½ m= n= cos tan γ ( D 3 n {cos( φ y πm ( + φ, M m n m, n }, where G is the fractal roughness that affects the amplification of the surface peaks an valleys (inepenent of the frequency, D is the fractal imension that etermines the contribution of the high an low frequency components on the surface (high value of D inicates that high-frequency components ominate over low frequency components, M is the number of superpose riges that are use to construct the surface, γ is a scaling parameter (a goo value is γ =.5, an ø m,n is a ranom phase. Furthermore, L is the length of the sample profile, n is the frequency ine (n ma = log(l/l s /logγ an n ma is an integer, an L s is the cut-off length referring to the resolution at which the asperities can be etecte. Values for the material epenent parameters can be foun in the literature. We have use values for LIGA nickel []: D =.485, G =.35 nm, L s =.47 nm, M = 0. In our case, the lengths of the small sies of a right triangle are µm, thus L = µm. Fig. gives eamples of a fractal surface an a surface that is generate using sinusoial functions. For a probabilistic Gaussian moel [], the probability ensity function ø(z of the roughness peak height z is (9 ( z / φ( z = e σ, (0 σ π where σ is the stanar eviation of the asperity heights. The height z is perpenicular to the mean line of the surface as shown in Fig. 3 (applies to other roughness moels as well. The implementation of surface roughness moels with parameterize surfaces is eluciate in Fig. 3 for a twoimensional case. For smooth surfaces, the vector between any two points l, k taken from the surfaces, is the ifference between the points, i.e., = l k. In case of rough surfaces, also the height of the roughness peak at each point has to be taken into consieration. The roughness peak height is obtaine from equations (8 (0, or from any other appropriate moel. The irection of the roughness peak Fig.. Eamples of rough triangle surfaces: a fractal surface (upper an a surface that is create by using sinusoial functions (lower. is then etermine by the unit-length normal n u of the surface. Thus, the vector from point k on the rough surface (i.e., triangle in this case to point l on the smooth surface (triangle becomes = z. ( l k nu For the sinusoial an the fractal roughness moels, the force calculation between rough surfaces requires the aition of asperity heights in the istance calculations (i.e., in the calculation of, as shown in (. After this aition, equation (4 can be use for the force evaluations. In case of the probabilistic moel, the probability ensity function has to be inclue in the calculations. Accoring to [], the van er Waals force is calculate as follows h ε F = A f ( h z φ( z z, ( where ε is the equilibrium istance for flat surfaces at contact (e.g., ε = 0. nm, h is the separation istance between a smooth surface an the mean line of a rough surface (see Fig. 3b, f is the close form solution for van er Waals force between two infinite parallel planes (force for a unit area, an A is the contact area of the interacting surfaces. In ( the van er Waals force is solve analytically for the smooth case an thus, the force calculation requires numerical computations only in one imension. In our case, the freeom in object orientation requires that also the force between smooth surfaces is calculate numerically. Therefore in our calculations, the f in ( is replace with equation (4 an since the area is implicitly inclue in (4, the factor A in ( is remove. Although not implemente in here, the moel in ( can be augmente to take into consieration also the spatial istribution of the roughness peaks []. IV. FORCE RESULTS In the following, results that are obtaine with ifferent roughness moels are presente. In each case, the interacting objects are two right triangles with two ientical small sies that have length of µm. The Hamaker constant use in the 9 calculations is 0 J. Unless otherwise state, the numerical integration is carrie out by using a lattice metho [8]. = l k l = l k n u z n u 3 z mean line k k a b Fig. 3. A vector in D that connects two points taken from the interacting surfaces. a Two smooth surfaces an b a smooth surface an a rough surface. The otte line in 3b represents the mean surface. l ma. asperity height h at 3 564

4 A. Sinusoial surfaces Fig. 4 epicts van er Waals interaction force as a function of the separation istance between a smooth triangle an a parallel rough triangle. The rough surface topography is generate with sinusoial functions accoring to (8. The separation istance h between the smooth surface an the mean line of the rough surface is efine as h = + a, where a is the amplitue of the roughness peak (i.e., the maimum height of the peak above the mean line as shown in Fig. 4. The results are given for three ifferent peak heights a ( nm, 0 nm, an 00 nm an for two ifferent peak ensities (4 an 40 perios on each small sie. For comparison, also the results for smooth surfaces are shown both analytical an numerical. The analytical results are obtaine by applying (6 for same size area as the interacting area between the triangles. This result is not irectly comparable since it gives the force for certain area taken from infinite planes an in case of triangles the interacting areas are obviously finite. The ifference is emphasize at larger separation istances. It is shown that surface roughness has a rather ramatic effect on the force values. At the separation istance of nm an with roughness peak heights of 00 nm (peak ensity is 40, the force value is almost 6000 times bigger for smooth surfaces than for a smooth surface interacting with a rough one. With a lower roughness peak height of 0 nm, the force is still over 400 times bigger for smooth surfaces than for a smooth an a rough surface. Also the roughness peak ensity has an effect on the force values. With fewer roughness peaks, the area that is at minimum istance from the smooth surface is smaller than with higher peak ensity, thus lowering the van er Waals force value. The results also inicate that the effect of roughness loses its effect at a point where the separation istance becomes larger than ten times the value of the asperity height (i.e., force values for rough surfaces approach the force values for smooth surfaces. Most of the results presente in this paper are obtaine for a smooth surface interacting with a rough one. This is base h Fig. 4. Van er Waals force as a function of the separation istance. A smooth triangle interacting with a rough one. Effects of roughness peak ensity an peak height on the force (sinusoial roughness peaks. For comparison, the force values are also presente for smooth surfaces (analytical an numerical results. α Fig. 5. Van er Waals force as a function of the inclination angle α. A smooth triangle interacting with a rough one. Effect of roughness peak height an separation istance on the force (sinusoial roughness peaks. For comparison, the force values are also presente for two smooth surfaces (numerical results. on the iea that two rough surfaces can be presente as a smooth surface an an equivalent rough surface. The variance an the mean of the equivalent surface are equal to the sums of the means an variances of the actual rough surfaces []. For comparison, Fig. 4 gives an eample of a situation where both interacting surfaces are rough. In Fig. 5, the van er Waals force is presente as a function of the inclination angle α when one of the surfaces is tilte. Tilting takes place relative to one of the small eges. The results are presente for two ifferent separation istances ( nm, 00 nm an for two ifferent roughness peak heights a ( nm, 0 nm. For comparison, also the force results between two smooth triangles are presente. B. Fractal surfaces Tilting with respect to one of the small sies Eamples of force results as a function of the separation istance for fractal surfaces are presente in Fig. 6. In the calculations, values for LIGA nickel are use as the material epenent parameters (see Section III. With ranom surfaces, the etermination of the separation istance in the same way as for the eterministic surfaces in Section IV.A prouces lower force values (for ientical values of. This is ue to the fact that on a ranom surface, all the peak heights o not have the same maimum value an the asperities are not evenly istribute on the surface. In Fig. 6, the results are given using three ifferent approaches to calculate the separation istance. In each case, the mean value of the roughness peaks is zero. In other wors, at zero lies the smooth reference surface an the asperity heights are ae to this value. The three ifferent approaches to calculate the istance are: istance between the smooth surface an the mean of the rough surface is h = + σ, where σ is the stanar eviation of the roughness peaks, istance between the smooth surface an the mean of the rough surface is h = + σ, an 3 the separation istance between the smooth surface an the mean of the rough surface is h = + z ma, where z ma refers to the maimum roughness peak height. In Fig. 6, the values 565

5 h P r n S P = V 0 + E u + E v S: (r V 0 (E E = 0 r is an intersection point, thus: r = P + t (E E an (r V 0 (E E = 0 σ case case case 3 σ zma mean line Fig. 7. At point P on triangle, the maimum height h of the roughness peak is limite by the surface S spanne by the eges of triangle. In case of non-conve surfaces, also the neighboring triangles affect h. h = r P Fig. 6. Van er Waals force as a function of the separation istance for parallel surfaces. A smooth triangle interacting with a rough one (fractal surface. Separation istance between the surfaces is etermine in three ifferent ways. Symbol σ refers to the stanar eviation of the roughness peak heights. for σ an z ma are m an m, respectively. Ecept for the thir case presente above, the metho to calculate the separation istance means that parts of the interacting surfaces are in contact as shown in Fig. 6. The calculation of the vector in ( oes not prevent the roughness peaks from penetrating the interacting surfaces. Thus, the contact situation has to be consiere separately. In our approach, the height of the roughness peaks that ecee the separation istance between the interacting surfaces is fie to a value h ε, where h is the separation istance between the surfaces at that point (between the smooth surface an the mean line of the rough surface, see Fig. 3b an Fig. 7 an ε is the interatomic equilibrium istance (here, ε = 0. nm. In other wors, in contact, the highest peaks are cut by the smooth surface. At those points, the force has its maimum value (force at the equilibrium istance. For parallel surfaces, the etermination of the separation istance h between them is trivial. For arbitrary orientations, when a rough surface is interacting with a smooth one, the calculation of h is one as follows V ( E E P ( E E ( E E ( E E 0 h =, (3 where V 0 is a verte point on triangle, P is a point on E E is the surface normal of triangle i, an triangle, ( i i is a norm that gives the length of the vector. The istance calculation is presente in Fig. 7. The metho is only suite for a rough surface interacting with a smooth one. In case of two rough surfaces, the istance evaluation epens on more than two points. Thus, also the neighboring points have to be taken into consieration. C. D. Probabilistic surfaces In Fig. 8, results obtaine with the probabilistic moel an Monte Carlo integration [] are presente. Force results are mean line Fig. 8. Surface roughness as a stochastic process. Probabilistic moel (Gaussian, where the peak height is a ranom variable. Hariri-moel [] is intene for infinite flat surfaces at close istances (numerical integration in only one imension, see (. shown as a function of the separation istance for parallel surfaces an for a situation where the smooth surface is tilte. Separation istance is calculate between the smooth surface an the mean line of the rough surface. The results are compare to those obtaine by using ( for two parallel infinite planes (i.e., numerical integration is implemente only in one imension since the van er Waals force for parallel infinite planes can be evaluate in close form. In all of the simulations, the equilibrium istance ε is 0. nm an the stanar eviation σ is 0.35 nm, which is the value for gol surface as given in []. The penetration of the roughness peaks to the interacting surface is etecte in a similar way as for the fractal surfaces, e.g., using (3. V. DISCUSSION Many engineering surfaces can be consiere as fractal (see, e.g., [] which justifies the use of fractal surface moel in van er Waals force evaluation. For the probabilistic moel, other peak height istributions besies Gaussian can be use. This augments the applicability of the moel for many types of rough surfaces. The roughness moel that utilizes sinusoial functions has low resemblance with real engineering surfaces. However, it gives straightforwar means to eamine the effect of roughness peak height an spatial peak ensity on the force values. Common problem with all the three roughness moels is 566

6 the high computational compleity that is involve in the force evaluation. This is cause by the highly nonlinear integran combine with the comple surface topography. The numerical integration for sinusoial an fractal surfaces was implemente with a lattice metho [8], that use approimately function evaluations. The fractal moel is computationally more emaning than the sinusoial moel since the (rough surface formation is more complicate an the penetration of roughness peaks to the interacting surface has to be separately etecte (applies to the probabilistic moel as well. In case of the probabilistic moel, computational compleity is further increase by the aitional (fifth integration imension that is require when compare to the other two methos. The lattice metho with function evaluations i not prouce reliable results. Thus, the results in Section IV.C were obtaine with Monte Carlo integration [3] using up to 0 9 function evaluations. The computational compleity makes the probabilistic moel inefficient for larger objects (when the imensions are measure in hunres, or even tens, of micrometers. An avantage of the probabilistic moel is that there is no nee to separately generate the rough topography uring force calculations. Therefore, the shape of the interacting surfaces or the selection of the function evaluation points cannot affect the force results in the same way as with the sinusoial or the fractal moel. The presente force calculation metho inepenent of the use surface roughness moel is especially suite for surfaces that can be ivie into flat sub-surfaces. For curve surfaces, the triangular surface approimation leas to errors as iscusse in Section II. This problem can be overcome with ifferent surface approimations, e.g., with splines [8]. Better applicability of the propose metho for micromanipulation purposes requires further research efforts at least in the following four areas: reuction of the computational compleity, better force moels, e.g., in terms of the retaration effect (here, the Hamaker s approach was obtaine in orer to have analytical reference results the metho in itself is not restricte to Hamaker s approach, 3 valiation of the results with measurement ata, an 4 combination of the force values with those representing capillary force an electrostatic force in orer to get the total force values. VI. CONCLUSIONS This paper presente approaches to calculate van er Waals force between two objects that have comple shapes, rough surfaces, an arbitrary orientations relative to each other. The surfaces of the interacting objects were approimate with triangular meshes an three ifferent surface roughness moels: sinusoial, fractal, an probabilistic. The force results showe that alreay small asperities an small inclination angles can consierably ecrease the effect of van er Waals force in object interaction. The results also inicate that beyon a certain separation istance, the roughness has no effect on the force values. Furthermore, the asperity peak height seems to have a greater impact on force values than the peak ensity. REFERENCES [] M. Maou. "Scaling laws, actuators, an power in miniaturization," in Funamentals of Microfabrication, M. Mark, E. Boca Raton: CLC Press, 997, pp [] W. S. N. Trimmer, "Microrobots an micromechanical systems," Sensors an Actuators, vol. 9, no. 3, pp , 989. [3] R. A. Bowling. "A theoretical review of particle ahesion," in Particles on Surfaces I: Detection, Ahesion an Removal, K. L. Mittal, E. New York: Plenum Press, 988, pp [4] F. Arai, D. Ano, T. Fukua, Y. Nonoa, an T. Oota, "Micro manipulation base on micro physics strategy base on attractive force reuction an stress measurement," in Proc. 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