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Simulation of micro-manipulations: Adhesion forces and specific dynamic models
International Journal of Adhesion & Adhesives 19 (1999) 35—48
Simulation of micro-manipulations: Adhesion forces and specific dynamic models Yves Rollot*, Ste´phane Re´gnier, Jean-Claude Guinot Laboratoire de Robotique de Paris, Universite& Paris 6 - Universite& Versailles Saint-Quentin-en-Yvelines, CNRS URA 1778, 10-12 Avenue de l+Europe, 78140 Ve& lizy, France Accepted 4 September 1998
Abstract Aiming at achieving precise micromanipulations using only adhesion forces, the three main sticking effects on a microscale are considered. On the hypothesis that Newton’s laws are applicable at this scale, dynamic models of a simple task, consisting in picking up and placing micro-spheres, are proposed. Three models of capture are written, introducing Van der Waals, capillary, electrostatic, rubbing and pull-off forces. A complete model of the release is also proposed. The capture and release tasks were dissociated and simulated. Some strong conclusions guaranteeing the manipulation by adhesion were extracted on materials of each part of the system, on the size of the spheres and on the speed limits applied to the end-effector. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Micro-scale; Adhesion; Dynamical model; Manipulation
1. Introduction Micro-robotics is a complementary way of research and is independent of classical robotics. Decreasing the scale brings us to technological limits such that it seems to be necessary to use intrinsic properties of the considered scale. For example, micro-manipulation totally differs from the classical prehension. Thus, the weight is generally negligible for a microscopic size system, making the classical systems useless and unusable. It is a matter of exploiting physical phenomena present at this scale (Van der Waals forces, capillary forces and electrostatic) and developing a new manipulation mode. This micromanipulation could thus be achieved by adhesion. Nevertheless, for exploiting and taking advantage of these particularities, it is useful to develop a dynamical model permitting us to theoretically validate the manipulation by adhesion, to define its main characteristics and to establish the necessary conditions of its existence. The topic of this paper is to establish this model and the
* Corresponding author. Tel.: #33 1 39 25 49 76; fax: #33 1 39 25 49 67.
different associated conclusions using a canonical example: the pick up and release of micro-spheres. In the first section the problematic of the chosen example is presented. The dynamical model is then introduced by considering the different forces successively. Simulations were carried out to validate the micromanipulation mode and results are presented. Some precise studies lead us to establish all the particularities of this manipulation.
2. Problematic Micro-robotics, and more specifically micro-manipulations are the main subject of this work. That is, manipulating objects of few cubic micrometers in size, and working with distances of the order of micrometers and submicrometers. The aim is to establish a general model of manipulation using characteristic forces at the microscale. To understand the different stages, a classical example of micro-spheres manipulation has been chosen. Dynamical modelling is thus achieved using this canonical example where the main particularity is to illustrate the main characteristics of the problem. The aim is to unpile a stack of 5 lm radius balls located on a substrate and to align them on the same
0143-7496/99/$—see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 7 4 9 6 ( 9 8 ) 0 0 0 5 5 - 4
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Y. Rollot et al. / International Journal of Adhesion & Adhesives 19 (1999) 35—48
scale [11]. Based on this hypothesis, modelling equations are proposed using the intersurface distances illustrated in Fig. 2a. At the starting point, the subtrate, the balls and the probe are in contact at D "0.4 nm (atomic contact distance):
Fig. 1. Schematic presentation of the manipulation.
substrate (a low radius value was chosen to characterize the manipulation on very little objects). To perform this manipulation by adhesion a micro-probe is used. Numerical data on the constitutive materials are summarized in the appendix [1]. Fig. 1 simplifies the different parts of the system and the stages of the process. A lot of parameters take place in the problem. Indeed, dynamical modelling must allow us to study material combinations, speed of the probe for the capture and the release, tilting angle of the probe for this release, etc. The considered forces are of different nature: Van der Waals [2,3], capillary [4,5] and electrostatic. Charge quantities at the microscopic scale are so little (around 10\ C/m) that triboelectric phenomena (or contact electrification) cannot be neglected [6]. In this application, all parts of the system are initially supposed to be free of charges. Only charges generated by triboelectrification are taken into account. Deformations of contacting surfaces are introduced on the hypothesis that the manipulation is rapid enough to neglect deformations on the ball—probe interface for the capture task, and at the ball—substrate interface for the release task. Refs [7,8] argued that for hard solids of small radius and low surface energy, the DMT theory [9] would be appropriate. This model has been chosen for the pull-off forces. Static rubbing forces [10] were also added in the case of a manipulation in a dry environment.
3. Dynamic models In this section, different models of probe and ball motion are presented. First, only Van der Waals forces and deformations are taken into account. Then, capillary, electrostatic and rubbing forces are added. 3.1. Dynamic model considering Van der Waals and pull-off forces This work is based on the hypothesis that Newtonian mechanics and Newton’s laws are applicable on a micro-
D (0)"D (0)"D (0)"D The different equations related to the system’s dynamics are obtained by writing the dynamic equilibrium of each part of the system (probe and spheres). A dynamic model is then written by integrating first only Van der Waals (VdW) forces. ‘‘Pull-off’’ forces will appear in the constraints. A R m ½= "F ! !m g 6D
(1)
(3 (3A R A R !m g m (D= # D= )" ! 2 6D 12D
(2)
A R (3A R ! !m g (3) m D= " 6D 24D where A , A and A are, respectively, the Hamaker constants for ball—probe, ball—substrate and ball—ball interfaces. A time law is applied to the speed ½Q of the probe (ramps or steps). The problem possesses four unknowns D , D , D and F . Another equation, coming from the geometric constraint is then added to the model (3 ½ "D #2R # (D #2R )#D . 2
(4)
The equations do not have the same importance in the system’s physical point of view. Only the last equation and its kinetic and dynamic derivatives have to be verified all the time. Physical compatibility tests have to be added to the three others. Thus, for the equation related to balls 2 and 3 (Eq. (3)), a condition on D= is always imposed by the substrate reaction (when the balls contact the substrate at D "D "0.4 nm): if D "0.4 nm N D= 50 constraint (1) In the same way, for Eq. (2): if D "0.4 nm N D= 50 constraint (2) The first equation of the dynamic model (Eq. (1)) is always verified because the constraint comes from the second derivative of the geometric relation.
The problem is supposed to be symmetric.
Y. Rollot et al. / International Journal of Adhesion & Adhesives 19 (1999) 35—48
37
Fig. 2. Distance notations for the dynamic models: (a) capture, (b) release.
Two discontinuities generated by the deformations are added to the previous constraints. An increase in ball—ball and ball—substrate contact surface, causing elastic flattening, generates pull-off forces. These forces are proportional to the work of adhesion ¼ [12] and the contact area. Using the DMT model of deformation, the detachment constraints have the following expressions:
To simplify the problem, the hypothesis for the calculation of the force are d"2r and h"0. Thus, capil lary forces are expressed by
F 'R n¼ and F '2R n¼ \ \ respectively for ball—ball and ball—substrate detachment. These expressions have to be taken as lower thresholds. ¼ is the energy which has to be given to separate isothermically in a reversible manner two solids by propagating the crack at the interface at zero speed. At the macroscopic scale, the magnitude of the pull-off forces is increased by the motion’s kinetics. At the microscopic scale, the influence of the speed is not exactly known. The forces may probably be increased. The influence law can only be experimentally extracted. Thus, these constraints coming from a purely thermodynamical argument represent a lower limit.
respectively, for the probe—ball, ball—substrate and ball—ball interactions. Initially, the different parts of the system are supposed to be free of charge. First, a conductor—insulator triboelectric phenomenon appears between substrate and balls [14]. »"(
!
)/e is the contact potential and » (0. The surface charge acquired by each material is [6]:
3.2. Dynamic model considering VdW, capillary and electrostatic forces Before writing the full expression of the model, it is necessary to express the capillary forces and to calculate the different electrostatic charges acquired by triboelectrification. Humidity is arbitrarily fixed at 50%. The radius of a toric concave meniscus between contacting surfaces is then approximated to r "1.6 nm [13]. This menis cus generates capillary forces because of Laplace pressure. For a flat-sphere interaction, it has the following expression: F "4nRc cos h/(1#D/d) with J h liquid/surface contact angle d immersion height and D contact distance
4nc R d 2nc R d 4nc R d F " J , F " J , F " J d#D d#D d#D (5)
p"#+2e e (n !n )e» , P ' where n and n are the density of empty and full traps ' and e the dielectric constant of the insulator. P The total charge Q acquired by the ball and the insulator is then Q "p4nR r where 4nR r is the contact area and r the
radius of the aqueous meniscus between the contacting surfaces. Each ball possesses, after triboelectrification, this charge Q . The probe then contacts ball 1. A conduc tor—conductor triboelectric phenomenon appears. Ball 1 possesses the charge Q , which is supposed to be concentrated at the center of the sphere. Its surface potential is » "Q /(4ne e R ) 1 P where e is the dielectric constant of the medium (in this P case silicon). The work function of the ball is: "
#e» 1 [15]. The contact potential generated by the probe—ball contact is »"( ! )/e and the charge acquired by each part is Q "e A» /z
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Y. Rollot et al. / International Journal of Adhesion & Adhesives 19 (1999) 35—48
with z "100 nm [16] and A"4nR r . Thus, the total charge on the balls is Q"Q #Q Each part of the system is then charged giving rise to Coulomb forces in the dynamic model: m ½= "F !F45!F !F !m g
(6)
m ½= "F45#F #F !(3F45 !(3F !(3F !m g
(7)
(3 (3 (3 F # F m D= " F45# 2 2 2
!F45!F !F !m g
(8)
(3 ½ "D #2R # (D #2R )#D 2
(9)
Electrostatic forces are discontinuous. They only exist when the separation distances D become greater than the G electron tunnelling distance D "1 nm [17]. These forces have the following expressions: F "0 if D (D , Q F " if D 5D , 4ne D F "0 if D (D #d , Q if D 5D #d , F " @ 4ne D F "0 if D (D #d QQ if D 5D #d F " 4ne D where d and d are the central displacements generated by the ball—ball and ball—substrate contact. These displacements are computed with expressions of the JKR model of deformation [18]. d equals zero because the manipulation is supposed to be rapid enough to cause no deformation on this contact. Constraints (1) and (2) expressed in the previous model are the same in this model. Only detachment constraints are different. Because of condensed water on the surfaces, the work of adhesion is changed: F 'R n¼ and \ \ F '2R n¼ \ \ 3.3. Dynamic model with friction Rubbing appears in equilibrium when manipulating in a dry environment. In a humid atmosphere, the conden-
sed water on the surfaces acts as a good lubricant and the rubbing coefficient k equals zero. Expression of the rubbing force is F"k(F #F ) [10] (where F is the external load). It opposes the motion and is always perpendicular to the adhesion force. Dynamic equations are m ½= "F !F45!F !m g N m ½= "F45#F !(3F45 !(3F !kF45!m g @ (3 (3 m D= " F45# F @ 2 2
(10)
(11)
!F4B5!F !m g (12) @ The rubbing force is discontinuous. It exists if D (D #d . For metals, static rubbing coefficient is in the range 0.15—0.3 Ref. [10] gives a value of k"0.2 for an iron—silicon contact. In a first approximation, this value is used. Discontinuity in the forces are expressed by the variations of k:
0.2 if D 4D #d 0 if D 'D #d Constraints and detachment conditions are the same as in the first model (VdW only). k"
3.4. Dynamic model of the release task For the release task, a complete model integrating Van der Waals, capillary, electrostatic and pull-off forces is presented. The task is outlined in Fig. 2b. The same discontinuities exist. The model is m ½= "F !F45 cos(h)!F cos(h) !F cos(h)!m g (13) m D= "F45 cos(h)#F cos(h)!F45 !F #F cos(h)!F !m g (14) ½Q "DQ #DQ cos(h) (15) where ½ "D #R #(R #D ) cos(h) and ½= "D= #D= cos(h). The substrate is supposed to be free of charge before the arrival of the ball. An electrostatic force called ‘‘image force’’ arises. The ball and the probe are still charged with Q. The image force will then arise if D 'D : F "0 if D (D , Q (e !1) 1G if D 5D , F " 16ne D (e #1) 1G F "0 if D (D Q F " if D 5D #d 4ne D
Y. Rollot et al. / International Journal of Adhesion & Adhesives 19 (1999) 35—48
Manipulation time is short enough to neglect ball—substrate deformation. Nevertheless, a detachment condition exists in the ball—probe contact because of deformations caused by adhesion forces. This constraint is
39
4.1. The capture task (VdW and pull-off forces) In this case of simulation, the calculated values of pull-off forces are
4. Simulations, results and discussion
F "1.24;10\ N and \ F "7.62;10\ N \ The first capture occured in the first approximation when
The capture task models previously described were implemented with the Matlab—Simulink tool. Fig. 3 shows the graph which has been used. The aim is to dertermine a range of probe displacement speed ensuring capture by adhesion. The result of the manipulation appears on the curves showing the evolution of the three characteristic distances of the problem vs time (D ball—substrate distance, D ball—ball distance and D ball—probe distance). The capture is efficient when D increases and D "D . A time law is applied to the speed ½Q of the probe (ramps or steps). During the simu lations, the speed initially equals zero. The fixed final speed (or step value) is reached in 1 ns.
F 'F (16) \ During manipulation simulations, the speed limit for the first capture is determined between the step values 4;10\ m s\ (or a corresponding external force of 7.5;10\ N) and 5;10\ m s\ (or a corresponding external force of F "9;10\ N) (Fig. 4). If the step value is increased, a value at which no capture occurs is reached. By simulation, this value is determined between the step values 9;10\ m s\ (F "0.0163 N) and 1;10\ m s\ (F "0.0182 N) (Figs. 5—7). When manipulation tasks are based only on Van der Waals forces and deformations, the displacement speed of the effector dominates. Simulation results
-
F '2R n¼ cos(h) \ \
-
Fig. 3. Simulink graph for the capture task.
Fig. 4. Distances vs. time for the transition between no-capture (4;10\ m s\) and capture (5;10\ m s\).
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Y. Rollot et al. / International Journal of Adhesion & Adhesives 19 (1999) 35—48
Fig. 5. Evolution of distances for the transition capture (9;10\ m s\) and no-capture (1;10\ m s\).
Fig. 6. Evolution of ball—ball and ball—probe VdW forces for step value 9;10\ m s\.
Fig. 7. Evolution of ball—ball and ball—probe VdW forces for step value 1;10\ m s\.
show a minimum and a maximum speed of capture. As the fixed final speed value is established in 1 ns, those minimum and maximum values correspond to minimum and maximum acceleration values transmitted to the object to be manipulated by adhesion. We thus find a range of accelerations allowing the manipulation by
adhesion. By increasing the step value between the minimum and the maximum values, distance evolutions vs time are quite the same whatever forces are considered in the model. From the step value of first capture to the first step of no-capture, we obtain five profiles of distance evolutions shown in Fig. 8.
Y. Rollot et al. / International Journal of Adhesion & Adhesives 19 (1999) 35—48
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Fig. 8. Profiles of distances evolution.
When considering Van der Waals and pull-off forces, the fives characteristic stages can be drawn out as follows: (i) first capture F 'F , \ (ii) slight detachment between the ball and the probe with D (D : capture (3 (3 D= "½= ! D= '0 N ½= ' D= (0) 2 2
(iii) large detachment between the ball and the probe with D 'D : capture D= 'D= N ½= '(3D= (0) (iv) no capture: D is always greater than D , (v) no capture and no detachment between balls:
Stages (i) and (iv) can be considered as critical points corresponding to limit external forces applied to the probe. The lower limit comes from inequation (16), which, applied to Eq. (1) gives
(3
!
2
D= (1)*t!2
(3 R\ "D (0)#½= *t! D= (i)*t 2 G
(18)
In the same way, for D (t): R\ D (t)"D (0)# D= (i) *t G
(19)
k º" ( L (D #½= *t! L\ º *t) G G k ! (D # L\ º *t) G G
(20)
The series S associated to º is defined by S " L\ º . L L L G G Expression (20) then becomes
A R m ½= # #m g5F 6D
k º" L ( (D #½= *t! S *t) L\
1 A R N ½= 5 F - ! m 6D
where D "4;10\ m and ½Q "½= *t. The upper limit depends on more complex relationships. To evaluate this limit, Eq. (2) is written as follows: k k D= (t)" ! D (t) D (t)
(3 D (t)"D (0)# ½= ! D= (0) *t 2
Replacing expressions (18) and (19) in Eq. (17), a series is obtained:
D= (1)"!D= (0)
equation:
(17)
where k "2.67;10\, k "1.78;10\, D= (0) "5.59;10 m s\ and D (0)"D (0)"D . Considering the speed profile of the probe to be a step, D at instant t can be expressed by the following
k ! (D #S *t) L\
(21)
The aim is to define the upper limit of the external force ensuring the capture. This is obtained when D (t) and D= decrease, i.e.: º 'º ∀n. L L> This series cannot be written as a recurring series. The problem can be summarized in three points: z ½= is the unknown of the problem, z º is a descending series, L z S is a growing series, then º (S ((n!1)º L L\
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Y. Rollot et al. / International Journal of Adhesion & Adhesives 19 (1999) 35—48
using Eq. (21) k º' L ( (D #½= *t! S *t!º *t) L\ L k ! (22) (D #S *t#º *t) L\ L The resolution of this last equation with MapleV gives two results but only one compatible with the problem. The corresponding solution is ½$ *t'0.92;10\#0.22;10\S L\ The minimum value is then obtained by reducing S and replacing it by º . The upper speed limit L\ ensuring the capture is then ½Q "9.31;10\ m s\ By simulation, this speed limit has been found to be » +9.34;10\ m s\.
4.2. The capture task (VdW, capillary, electrostatic and pull-off forces) For this model, contact electrification and capillary phenomena are added. Humidity is fixed at 50%
(r "1.6 nm). The calculated values of pull-off forces
are: F "6.67;10\ N and \ F "3.72;10\ N \ The lowering of the pull-off force magnitudes (compared to previous model) is due to the condensed water on surfaces which decreases adhesion work. These pull-off forces are weak compared to the capillary force magnitudes. In this case, the external force applied to the probe is always greater than the pull-off forces (Eq. (1) gives the minimum value for the external force equal to 5.2;10\N). Ball 1 is then captured at a weak value of ½Q . There is no minimum critical time law for the capture. Experimentally, the detachment kinetic may increase the magnitude of the pull-off forces and thus a lower critical speed limit may arise reducing the admissible speed range. When the step value is increased, a value at which no capture occurs is reached. By simulation, this limit is found to be between the values 2.3;10\ m s\ and 2.4;10\ m s\ (Fig. 9). Figs. 10 and 11 show the evolution of VdW, capillary and electrostatic forces vs time for both interactions, ball—ball and ball—probe and the twostep value limits.
Fig. 9. Evolution of distances for the transition between capture (2.3;10\ m s\) and no-capture (2.4;10\ m s\).
Fig. 10. Evolution of ball—ball forces for step values 2.3;10\ and 2.4;10\ m s\.
Y. Rollot et al. / International Journal of Adhesion & Adhesives 19 (1999) 35—48
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Fig. 11. Evolution of ball—probe forces for step values 2.3;10\ and 2.4;10\ m s\.
For the step value 2.3;10\ m s\, the appearance of the ball-probe electrostatic force between instants 20 and 30 ns causes the decreasing of D= . This is after all ob served in the D profile. The second derivative of geomet ric relation (4) forces D= to evolve in the opposite way. At time 60 ns, D becomes greater than the electron tunnel ling distance (D ) and the ball—ball electrostatic force arises (Fig. 10). This force, whose magnitude decreases rapidly is not strong enough to oppose ball—probe forces. As D is invariant between 60 and 120 ns, it allows D to linearly increase. Ball—ball forces are then too weak to oppose ball—probe forces and the capture occurs. For the step of value 2.4;10\ m s\, the probe-ball electrostatic force which appears (Fig. 11), increases D= . This phenomenon is observed on the D profile between instants 40 and 60 ns. D= decreases under the influence of the ball—ball electrostatic force. This decreasing, via Eq. (4), allows D to increase and the magnitudes of ball—probe forces to decrease rapidly. Ball—ball forces dominate the equilibrium of ball 1, bringing it into contact with balls 2 and 3. Capture does not occur.
If the step value is increased, an irreversible break in the probe—ball adhesion is reached. By simulation this limit has been found in the range 6.9;10\ —7;10\ m s\. This is smaller than the one found for the ‘‘VdW—pull-off’’ model. In the first instants of the manipulation, electrostatic forces are non-existent and the model is the same as the ‘‘VdW—pull-off’’ model except Eq. (11) related to D= . Rubbing forces decrease D= and because of the second derivative of Eq. (4) they increase D= . The gap between D and D increases with the magnitude of the applied step till the adhesion breaks definitively at 7;10\ m s\. Deformations are not negligible compared to the critical distance D (d +3.8 nm, d + \ \ 1 nm). Electrostatic forces then appear too late to oppose the increase of the gap and to permit the manipulation by adhesion.
4.3 The capture task (VdW, electrostatic, friction and pull-off forces)
As it has been shown previously, the probe’s speed plays an important parameter in the object capture. The following is consecrated to the influence of this speed on the release task. In order to achieve the release task, the probe is tilted at an angle h. ½Q and h are the problem’s variables. The task is realised in a humid environment. Thus, Van der Waals, capillary and electrostatic forces are taken into account. Manipulation time is short enough to ignore ball—substrate deformation. Nevertheless, the probe—ball deformation imposes a pull-off condition expressed by
Capillary forces do not appear in this model. The behaviour of the admissible manipulation speed thus follows the ‘‘Vdw—pull-off’’ model. This indicates that each force has comparable magnitude and confirms capillary forces predominance compared to electrostatic and VdW forces. The rubbing forces are only present on contact. Their influence is then limited to the primary instant of the manipulation. When contact is broken only VdW and electrostatic forces act. Rubbing coefficient equals 0.2. For this value, the first step of capture equals the one of the ‘‘VdW—pull-off’’ model: 5;10\ m s\. This step value changes if the rubbing coefficient is increased to 0.9. For lower values, rubbing does not play any role in the magnitude of the external force needed for the first capture.
4.4. The release task
F 'F cos(h) \ -
(23)
The probe is in gold (soft material) thus deformations are important. Calculation of d gives d "37.75;10\ m
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Y. Rollot et al. / International Journal of Adhesion & Adhesives 19 (1999) 35—48
Gold and Silicon possess a high Hamaker constant. Despite water in the interface, the pull-off force is important: F "9.3;10\ N. \ The release is efficient if D "D (ball—substrate dis tance) and if D '0 (probe—ball distance). The curves in this part will show the evolution of D , D and d . This last value will represent the evolution of the central displacement vs time. Distance D will equal zero if d'0. Simulations have been carried out. They show that because of capillary forces, very weak step values allow the release. Thus, a step of value 10\ m s\ combined with an angle h3[88°, 90°] makes the release efficient. As can be predicted with Eq. (23) the angle range increases with the value of the step applied. For example, this band is [76°, 90°] for a step of 10\ m s\. The specificity of these low speeds can be seen on the D profile. No bump is observed at any angle of the band imposed to the probe. The release speed limits vs angle h can be calculated with Eq. (23): m ½= k k k # # # cos h D 2r #D D
m g # 'F \ cos h
4.5. Conclusions on simulations
This inequation has to be verified when D "D . At this distance, the weight is negligible compared to other forces, and k , which represents the electrostatic force equals zero. Constants of the previous equation have then the following values: k "1.69;10\, k "1.47;10\,
the release. When the step value is increased, bumps on the D profile occur, indicating a breaking in ball—substrate contact. The magnitude of this detachment decreases when h increases but is always present for speeds greater than 10\ m s\. Distance characteristic profile evolutions vs. time are shown on the following curves for high applied speed. The applied speed is ½Q "10\ m s\ and imposed angles h"65°, 68° and 70°. Fig. 12b illustrates a case where the release does not occur. Figs. 13 and 14 show two cases of release. The relationship which exists between displacement speed and probe angle, in the aim of calculating the upper speed limit of release, is complicated. It is related to the discontinuity of electrostatic forces and accelerations ½= , D= and D= . Reasoning on the same basis as in the ‘‘VdW—pull-off’’ capture model could nevertheless be carried out. To ensure a precise release, it is important to avoid a bump on the D profile. For low displacement speeds (lower than 10\ m s\), breaking in the ball—substrate contact can be avoided by imposing a high value of h (greater than 80°). Choosing a low displacement speed and a high angle h guarantees a precise release.
k "0
Results are collected on the curve (Fig. 12a) that shows the minimum speed vs angle applied to guarantee
As it was explained in previous sections, the motion’s kinetics may act on the pull-off values and then change the speed limit values. These limits must be confirmed by a set of rigorous experiments. The model proposed here can be generalized for other manipulations and materials. Fig. 15 summarises the speed range limits applied to the system which ensure the capture by adhesion when different forces are considered. More details on the release are collected in Fig. 12a.
Fig. 12. (a) Lower limit for the release task as a function of h and speed, (b) Evolution of distances for a step value 10\ m s\ and h"65°.
Y. Rollot et al. / International Journal of Adhesion & Adhesives 19 (1999) 35—48
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Fig. 13. Evolution of distances for a step value 10\ m s\ and h"68°.
Fig. 14. Evolution of distances for a step value 10\ m s\ and h"70°.
Fig. 15. Summary of speed range ensuring capture by means of different force combinations.
5. Parameters sensitivity
5.1 Influence of materials combination
In this section, parameters of the system are more precisely studied. The studies have been carried out in the most simple case with the ‘‘VdW—pull-off’’ model. These studies pointed out useful information on the manipulation by adhesion.
Two inequations on Hamaker constants facilitating manipulation by adhesion have been extracted from a static consideration: A 'A
and
A '(3A /2
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Y. Rollot et al. / International Journal of Adhesion & Adhesives 19 (1999) 35—48
In previous sections, materials were chosen to respect these inequations. In the aim to observe the role of the interfaces, constitutive materials of different parts of the system have been changed. First, the following combination was chosen: z probe in polystyrene, balls in silicon, substrate in gold. This choice allowed no manipulation. Accelerations D= and D= are negative (Eqs. (2) and (3)), and distances D and D always equal the contact value D . In this case, adhesion probe-ball is too weak to oppose ball—ball and ball—substrate adhesions. This example indicates that when manipulating in a dry environment, a strong adhesion on the probe is needed. This implies that the probe must be made of a high Hamaker constant material. This conclusion is also valid in a humid environment. Indeed, for the two configurations of capture: z ¹he capture of ball 1: In this case, if D " \ D "D then F 'F eases the cap \ \ \ ture task. z ¹he capture of balls 2 and 3: In this case, if D \ "D "D then F "F \ \ \ keeps the predominence of F 45 in the capture \ task. A second combination has been simulated: z Probe in gold, balls in polystyrene, substrate in silicon. This choice guarantees a strong adhesion for the ball—probe interface and decreases the ball—substrate adhesion. The general behaviour during simulation is close to the one obtained with the gold—silicon—polystyrene combination. In fact, distance D is constant at the contact value D . This behaviour comes from Eq. (3). At contact, the weight is negligible. The equation is positive if: (3A
R A R ' (24) 24D 6D Hamaker’s combination relationships are supposed to be applicable. A can then be related to A and A : A "(A A 1 Putting in inequation (24): (3A
/4'(A A thus A '16A /3 With this condition, distance D increases, indicating that balls contacting the substrate are also captured. This hypothesis is not suitable for the task to be achieved. Simulation results then show comparable behaviour but characteristic values are shifted forward. The first capture occurs for a step value of 6;10\ m s\, and the last capture occurs for a step value of 2.1;10\ m s\. This shift is due to the decreasing of the ball—ball pull-off
force (1.93;10\ N vs 7.68;10\ N). The capture band is increased because of the lower limit variation. This points out the importance of the pull-off force as a condition for the manipulation. Computing the magnitude of this force depends on Hamaker’s constant and adhesion energies. These characterize the range of admissible manipulation speeds. Nevertheless, in a humid environment, the lower limit does not exist (in a first approximation, neglecting kinetic effects during detachment at the microscale). Variation on the band by changing the material combination is then very weak. The necessary condition ensuring manipulation by adhesion is the choice of a probe with a high Hamaker’s constant compared to the one of the balls and the substrate. 5.2 Influence of sphere radius This section is consecrated to the study of the influence of the sphere radius on the manipulation task. This study is always carried out with the ‘‘VdW—pull-off’’ model. Eqs. (1)—(4) can be written as follows: m ½= "F !C R !m g
(25)
(3 C C D= # D= " !C R 2
(26)
C C D= " !C R
(27)
(3 ½ "D #2R # (D #2R )#D 2
(28)
Influence of the sphere radius on the minimum speed of capture is obtained by applying the pull-off condition (16) as follows: R ½= 5 (n¼ !C ) where m \ C "A /(6D), D "4;10\ m and ½Q "½= *t This expression shows that the minimum acceleration (i.e. the minimum speed) is linearly related to R . The upper limit is extracted from the following equation:
1 k k ! D= (t)" R D (t) D (t)
(29)
where R"R /5;10\ is the radii ratio (the initial radius is R "5 lm), k and k are the variables expressed in the section concerning the simulation of the ‘‘VdW—pull-off’’ model, D (0)"D (0)"D and D= (0)"(1/R);5.594;10 m s\.
Y. Rollot et al. / International Journal of Adhesion & Adhesives 19 (1999) 35—48
47
Fig. 16. Speed limits ensuring capture vs sphere radius for the ‘‘VdW—pull-off ’’ model.
D at time t can be written as follows if the speed profile is a step: (3 R\ D (t)"D (0)#½= *t! D= (i)*t 2 G In the same way for D (t): R\ D (t)"D (0)# D= (i)*t G Replacing these two expressions in Eq. (29), a series is obtained:
k 1 º" L R (D #½= *t!(S *t) L\ k ! (30) (D #S *t) L\ The aim is to define the limit of the capture force. D (t) and D= must decrease, thus: º 'º ∀n L L> After simplifications, this equation is no more related to R. Solution obtained in section ‘‘dynamic models of the capture task considering only Van der Waals forces’’ is then still valid and can be expressed:
½= *t'0.919;10\#0.2206;10\S (31) L\ where S can be reduced to D= /R. L\ The growing of radius brings to a decreasing of the maximum admissible speed of capture. The speed of
capture range shrinks till it becomes nil. Evolutions obtained by simulation are summarized in Fig. 16. The maximum radius of a silicon sphere allowing the manipulation mode proposed when only VdW and pulloff forces are taken into account is then R +120 lm. Conclusion Within the framework of manipulation of micro-objects, a manipulation mode was established. It has been theoretically validated by simulations using a general dynamic model of micro-manipulation. Some strong conclusions have been extracted on material combinations, geometry and speed for the feasibility and existing conditions. The next step of this work will be to carry out, in a free environment, experiments associated to the example developed in this paper. Reflection on possible components of the system, and particularly on the part capable of generating the accelerations needed for the capture task have permitted a final glimpse of the system. Other applications of this system are foreseen, for example, a tool which determines the Hamaker constants of different materials, or the kinetic behaviour of the pull-off forces at the micro-scale.
Acknowledgements To M. Barquins, Research Director at the CNRS, for the fruitful discussion on the topic and to Sue McCarthy for her help.
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Y. Rollot et al. / International Journal of Adhesion & Adhesives 19 (1999) 35—48
References [1] Rollot Y, Regnier S, Guinot J-C. Microrobotique: mode`le dynamique et loi horaire pour une micromanipulation par adhe´sion. C. R. Acad Sci Paris Se´r II 326 b, 1998;469—74. [2] Hamaker HC. The london Van der Waals attraction between spherical particles. Physica 1937;10:1058—72. [3] Israelachvili JN. The nature of Van der Waals forces. Contemp Phys 1974;15(2):159—77. [4] Adams MJ, Perchard V. The cohesive forces between particles with interstitial liquid. Int Chem E. Symp Ser No. 91, 1963; 147—160. [5] Mastrangelo C. Mechanical stability and adhesion of microstructures under capillary forces — Part I: Basic theory. J Microelectromech Systems 1993;2(1):33—43. [6] Lowell, J, Akande A. Contact electrification — why it is variable?. J Phys D 1988;21:125—37. [7] Horn RG, Israelachvili JN, Pribac F. Measurement of the deformation and adhesion of solids in contact. J Colloid Interface Sci 1987;115(2):480—92. [8] Maugis D. Adhesion of spheres: the JKR-DMT transition using a dugdale model. J Colloid Interface Sci 1992;150(1): 243—69.
[9] Derjaguin BV, Muller VM, Toporov YU P. Effect of contact deformations on the adhesion of particles. J Colloid Interface Sci 1975;53(2):314—26. [10] Ando Y, Ishikawa Y, Kitahara T. Friction characteristics and adhesion force under low normal load. J Tribol 1995;117:569—74. [11] Feynman RP. There’s plenty of room at the bottom. Talk at the Annual Meeting of the American Physical Society at the California Institute of Technology (Caltech), December 29th, 1959. [12] Kaeble DH. Physical chemistry of adhesion. New York; Wiley Interscience Publication, 1971. [13] Israelachvili J. Intermolecular and surface forces. New York: Academic Press, 1991. [14] Lowell J, Rose-Innes AC. Contact electrification. Adv Phys 1980;29(6):947—1023. [15] Hays DA. Electrostatic adhesion of non-uniformly charged dielectric sphere. Int Phys Conf Ser. No. 118: Sec. 4, 1991: 223—8. [16] Lowell J, Akande A. Contact electrification of metals. J Phys D 1975;8:53—63. [17] Krupp H. Particle adhesion: theory and experiments. J Adv Colloid 1967;1:111—239. [18] Johnson KL, Kendall K, Roberts AD. Surface energy and the contact of elastic solids. Proc R Soc Lond 1971;A 324:301—13.
Appendix Numerical values for different Where: o Volumic weight (10 kg/m) e dielectric constant E electronic affinity (eV) c surface energy (mJ/m)
components K stiffness (;10 Pa) n refraction index E Ionization potential (eV) ' A Hamaker’s constant (10\ J)
M molar weight a polarizability (10\ cm)
+(E #E )/2 work function (eV) ' k viscosity
Materials
o
e
n
a
E '
E
c
A
K
M
Gold (Au) Polystyrene (PS) Silicon (Si)
19.283 1.05 2.33
— 2.55 11.7
— 1.55 3.41
5.8 15 5.38
5.1 4.9 4.85
9.23 8.43 8.15
2.31 1.37 1.385
1450 35.5 1400
54.6 7.29 25.8
2.5 80 140
196.97 104.15 28.0855
Water o
e
n
E '
a
M
k
0.9975
80
1.333
12.612
1.45
18.05
1.854
Hamaker constant’s for different contacts A 1
A
33.6
13.2
1 .1
A 5 1
A
20.3
13.8
1 5 1
A 1 5 .1 0.27
Simulation of micro-manipulations: Adhesion forces and specific dynamic models Yves Rollot*, Ste´phane Re´gnier, Jean-Claude Guinot Laboratoire de Robotique de Paris, Universite& Paris 6 - Universite& Versailles Saint-Quentin-en-Yvelines, CNRS URA 1778, 10-12 Avenue de l+Europe, 78140 Ve& lizy, France Accepted 4 September 1998
Abstract Aiming at achieving precise micromanipulations using only adhesion forces, the three main sticking effects on a microscale are considered. On the hypothesis that Newton’s laws are applicable at this scale, dynamic models of a simple task, consisting in picking up and placing micro-spheres, are proposed. Three models of capture are written, introducing Van der Waals, capillary, electrostatic, rubbing and pull-off forces. A complete model of the release is also proposed. The capture and release tasks were dissociated and simulated. Some strong conclusions guaranteeing the manipulation by adhesion were extracted on materials of each part of the system, on the size of the spheres and on the speed limits applied to the end-effector. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Micro-scale; Adhesion; Dynamical model; Manipulation
1. Introduction Micro-robotics is a complementary way of research and is independent of classical robotics. Decreasing the scale brings us to technological limits such that it seems to be necessary to use intrinsic properties of the considered scale. For example, micro-manipulation totally differs from the classical prehension. Thus, the weight is generally negligible for a microscopic size system, making the classical systems useless and unusable. It is a matter of exploiting physical phenomena present at this scale (Van der Waals forces, capillary forces and electrostatic) and developing a new manipulation mode. This micromanipulation could thus be achieved by adhesion. Nevertheless, for exploiting and taking advantage of these particularities, it is useful to develop a dynamical model permitting us to theoretically validate the manipulation by adhesion, to define its main characteristics and to establish the necessary conditions of its existence. The topic of this paper is to establish this model and the
* Corresponding author. Tel.: #33 1 39 25 49 76; fax: #33 1 39 25 49 67.
different associated conclusions using a canonical example: the pick up and release of micro-spheres. In the first section the problematic of the chosen example is presented. The dynamical model is then introduced by considering the different forces successively. Simulations were carried out to validate the micromanipulation mode and results are presented. Some precise studies lead us to establish all the particularities of this manipulation.
2. Problematic Micro-robotics, and more specifically micro-manipulations are the main subject of this work. That is, manipulating objects of few cubic micrometers in size, and working with distances of the order of micrometers and submicrometers. The aim is to establish a general model of manipulation using characteristic forces at the microscale. To understand the different stages, a classical example of micro-spheres manipulation has been chosen. Dynamical modelling is thus achieved using this canonical example where the main particularity is to illustrate the main characteristics of the problem. The aim is to unpile a stack of 5 lm radius balls located on a substrate and to align them on the same
0143-7496/99/$—see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 7 4 9 6 ( 9 8 ) 0 0 0 5 5 - 4
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Y. Rollot et al. / International Journal of Adhesion & Adhesives 19 (1999) 35—48
scale [11]. Based on this hypothesis, modelling equations are proposed using the intersurface distances illustrated in Fig. 2a. At the starting point, the subtrate, the balls and the probe are in contact at D "0.4 nm (atomic contact distance):
Fig. 1. Schematic presentation of the manipulation.
substrate (a low radius value was chosen to characterize the manipulation on very little objects). To perform this manipulation by adhesion a micro-probe is used. Numerical data on the constitutive materials are summarized in the appendix [1]. Fig. 1 simplifies the different parts of the system and the stages of the process. A lot of parameters take place in the problem. Indeed, dynamical modelling must allow us to study material combinations, speed of the probe for the capture and the release, tilting angle of the probe for this release, etc. The considered forces are of different nature: Van der Waals [2,3], capillary [4,5] and electrostatic. Charge quantities at the microscopic scale are so little (around 10\ C/m) that triboelectric phenomena (or contact electrification) cannot be neglected [6]. In this application, all parts of the system are initially supposed to be free of charges. Only charges generated by triboelectrification are taken into account. Deformations of contacting surfaces are introduced on the hypothesis that the manipulation is rapid enough to neglect deformations on the ball—probe interface for the capture task, and at the ball—substrate interface for the release task. Refs [7,8] argued that for hard solids of small radius and low surface energy, the DMT theory [9] would be appropriate. This model has been chosen for the pull-off forces. Static rubbing forces [10] were also added in the case of a manipulation in a dry environment.
3. Dynamic models In this section, different models of probe and ball motion are presented. First, only Van der Waals forces and deformations are taken into account. Then, capillary, electrostatic and rubbing forces are added. 3.1. Dynamic model considering Van der Waals and pull-off forces This work is based on the hypothesis that Newtonian mechanics and Newton’s laws are applicable on a micro-
D (0)"D (0)"D (0)"D The different equations related to the system’s dynamics are obtained by writing the dynamic equilibrium of each part of the system (probe and spheres). A dynamic model is then written by integrating first only Van der Waals (VdW) forces. ‘‘Pull-off’’ forces will appear in the constraints. A R m ½= "F ! !m g 6D
(1)
(3 (3A R A R !m g m (D= # D= )" ! 2 6D 12D
(2)
A R (3A R ! !m g (3) m D= " 6D 24D where A , A and A are, respectively, the Hamaker constants for ball—probe, ball—substrate and ball—ball interfaces. A time law is applied to the speed ½Q of the probe (ramps or steps). The problem possesses four unknowns D , D , D and F . Another equation, coming from the geometric constraint is then added to the model (3 ½ "D #2R # (D #2R )#D . 2
(4)
The equations do not have the same importance in the system’s physical point of view. Only the last equation and its kinetic and dynamic derivatives have to be verified all the time. Physical compatibility tests have to be added to the three others. Thus, for the equation related to balls 2 and 3 (Eq. (3)), a condition on D= is always imposed by the substrate reaction (when the balls contact the substrate at D "D "0.4 nm): if D "0.4 nm N D= 50 constraint (1) In the same way, for Eq. (2): if D "0.4 nm N D= 50 constraint (2) The first equation of the dynamic model (Eq. (1)) is always verified because the constraint comes from the second derivative of the geometric relation.
The problem is supposed to be symmetric.
Y. Rollot et al. / International Journal of Adhesion & Adhesives 19 (1999) 35—48
37
Fig. 2. Distance notations for the dynamic models: (a) capture, (b) release.
Two discontinuities generated by the deformations are added to the previous constraints. An increase in ball—ball and ball—substrate contact surface, causing elastic flattening, generates pull-off forces. These forces are proportional to the work of adhesion ¼ [12] and the contact area. Using the DMT model of deformation, the detachment constraints have the following expressions:
To simplify the problem, the hypothesis for the calculation of the force are d"2r and h"0. Thus, capil lary forces are expressed by
F 'R n¼ and F '2R n¼ \ \ respectively for ball—ball and ball—substrate detachment. These expressions have to be taken as lower thresholds. ¼ is the energy which has to be given to separate isothermically in a reversible manner two solids by propagating the crack at the interface at zero speed. At the macroscopic scale, the magnitude of the pull-off forces is increased by the motion’s kinetics. At the microscopic scale, the influence of the speed is not exactly known. The forces may probably be increased. The influence law can only be experimentally extracted. Thus, these constraints coming from a purely thermodynamical argument represent a lower limit.
respectively, for the probe—ball, ball—substrate and ball—ball interactions. Initially, the different parts of the system are supposed to be free of charge. First, a conductor—insulator triboelectric phenomenon appears between substrate and balls [14]. »"(
!
)/e is the contact potential and » (0. The surface charge acquired by each material is [6]:
3.2. Dynamic model considering VdW, capillary and electrostatic forces Before writing the full expression of the model, it is necessary to express the capillary forces and to calculate the different electrostatic charges acquired by triboelectrification. Humidity is arbitrarily fixed at 50%. The radius of a toric concave meniscus between contacting surfaces is then approximated to r "1.6 nm [13]. This menis cus generates capillary forces because of Laplace pressure. For a flat-sphere interaction, it has the following expression: F "4nRc cos h/(1#D/d) with J h liquid/surface contact angle d immersion height and D contact distance
4nc R d 2nc R d 4nc R d F " J , F " J , F " J d#D d#D d#D (5)
p"#+2e e (n !n )e» , P ' where n and n are the density of empty and full traps ' and e the dielectric constant of the insulator. P The total charge Q acquired by the ball and the insulator is then Q "p4nR r where 4nR r is the contact area and r the
radius of the aqueous meniscus between the contacting surfaces. Each ball possesses, after triboelectrification, this charge Q . The probe then contacts ball 1. A conduc tor—conductor triboelectric phenomenon appears. Ball 1 possesses the charge Q , which is supposed to be concentrated at the center of the sphere. Its surface potential is » "Q /(4ne e R ) 1 P where e is the dielectric constant of the medium (in this P case silicon). The work function of the ball is: "
#e» 1 [15]. The contact potential generated by the probe—ball contact is »"( ! )/e and the charge acquired by each part is Q "e A» /z
38
Y. Rollot et al. / International Journal of Adhesion & Adhesives 19 (1999) 35—48
with z "100 nm [16] and A"4nR r . Thus, the total charge on the balls is Q"Q #Q Each part of the system is then charged giving rise to Coulomb forces in the dynamic model: m ½= "F !F45!F !F !m g
(6)
m ½= "F45#F #F !(3F45 !(3F !(3F !m g
(7)
(3 (3 (3 F # F m D= " F45# 2 2 2
!F45!F !F !m g
(8)
(3 ½ "D #2R # (D #2R )#D 2
(9)
Electrostatic forces are discontinuous. They only exist when the separation distances D become greater than the G electron tunnelling distance D "1 nm [17]. These forces have the following expressions: F "0 if D (D , Q F " if D 5D , 4ne D F "0 if D (D #d , Q if D 5D #d , F " @ 4ne D F "0 if D (D #d QQ if D 5D #d F " 4ne D where d and d are the central displacements generated by the ball—ball and ball—substrate contact. These displacements are computed with expressions of the JKR model of deformation [18]. d equals zero because the manipulation is supposed to be rapid enough to cause no deformation on this contact. Constraints (1) and (2) expressed in the previous model are the same in this model. Only detachment constraints are different. Because of condensed water on the surfaces, the work of adhesion is changed: F 'R n¼ and \ \ F '2R n¼ \ \ 3.3. Dynamic model with friction Rubbing appears in equilibrium when manipulating in a dry environment. In a humid atmosphere, the conden-
sed water on the surfaces acts as a good lubricant and the rubbing coefficient k equals zero. Expression of the rubbing force is F"k(F #F ) [10] (where F is the external load). It opposes the motion and is always perpendicular to the adhesion force. Dynamic equations are m ½= "F !F45!F !m g N m ½= "F45#F !(3F45 !(3F !kF45!m g @ (3 (3 m D= " F45# F @ 2 2
(10)
(11)
!F4B5!F !m g (12) @ The rubbing force is discontinuous. It exists if D (D #d . For metals, static rubbing coefficient is in the range 0.15—0.3 Ref. [10] gives a value of k"0.2 for an iron—silicon contact. In a first approximation, this value is used. Discontinuity in the forces are expressed by the variations of k:
0.2 if D 4D #d 0 if D 'D #d Constraints and detachment conditions are the same as in the first model (VdW only). k"
3.4. Dynamic model of the release task For the release task, a complete model integrating Van der Waals, capillary, electrostatic and pull-off forces is presented. The task is outlined in Fig. 2b. The same discontinuities exist. The model is m ½= "F !F45 cos(h)!F cos(h) !F cos(h)!m g (13) m D= "F45 cos(h)#F cos(h)!F45 !F #F cos(h)!F !m g (14) ½Q "DQ #DQ cos(h) (15) where ½ "D #R #(R #D ) cos(h) and ½= "D= #D= cos(h). The substrate is supposed to be free of charge before the arrival of the ball. An electrostatic force called ‘‘image force’’ arises. The ball and the probe are still charged with Q. The image force will then arise if D 'D : F "0 if D (D , Q (e !1) 1G if D 5D , F " 16ne D (e #1) 1G F "0 if D (D Q F " if D 5D #d 4ne D
Y. Rollot et al. / International Journal of Adhesion & Adhesives 19 (1999) 35—48
Manipulation time is short enough to neglect ball—substrate deformation. Nevertheless, a detachment condition exists in the ball—probe contact because of deformations caused by adhesion forces. This constraint is
39
4.1. The capture task (VdW and pull-off forces) In this case of simulation, the calculated values of pull-off forces are
4. Simulations, results and discussion
F "1.24;10\ N and \ F "7.62;10\ N \ The first capture occured in the first approximation when
The capture task models previously described were implemented with the Matlab—Simulink tool. Fig. 3 shows the graph which has been used. The aim is to dertermine a range of probe displacement speed ensuring capture by adhesion. The result of the manipulation appears on the curves showing the evolution of the three characteristic distances of the problem vs time (D ball—substrate distance, D ball—ball distance and D ball—probe distance). The capture is efficient when D increases and D "D . A time law is applied to the speed ½Q of the probe (ramps or steps). During the simu lations, the speed initially equals zero. The fixed final speed (or step value) is reached in 1 ns.
F 'F (16) \ During manipulation simulations, the speed limit for the first capture is determined between the step values 4;10\ m s\ (or a corresponding external force of 7.5;10\ N) and 5;10\ m s\ (or a corresponding external force of F "9;10\ N) (Fig. 4). If the step value is increased, a value at which no capture occurs is reached. By simulation, this value is determined between the step values 9;10\ m s\ (F "0.0163 N) and 1;10\ m s\ (F "0.0182 N) (Figs. 5—7). When manipulation tasks are based only on Van der Waals forces and deformations, the displacement speed of the effector dominates. Simulation results
-
F '2R n¼ cos(h) \ \
-
Fig. 3. Simulink graph for the capture task.
Fig. 4. Distances vs. time for the transition between no-capture (4;10\ m s\) and capture (5;10\ m s\).
40
Y. Rollot et al. / International Journal of Adhesion & Adhesives 19 (1999) 35—48
Fig. 5. Evolution of distances for the transition capture (9;10\ m s\) and no-capture (1;10\ m s\).
Fig. 6. Evolution of ball—ball and ball—probe VdW forces for step value 9;10\ m s\.
Fig. 7. Evolution of ball—ball and ball—probe VdW forces for step value 1;10\ m s\.
show a minimum and a maximum speed of capture. As the fixed final speed value is established in 1 ns, those minimum and maximum values correspond to minimum and maximum acceleration values transmitted to the object to be manipulated by adhesion. We thus find a range of accelerations allowing the manipulation by
adhesion. By increasing the step value between the minimum and the maximum values, distance evolutions vs time are quite the same whatever forces are considered in the model. From the step value of first capture to the first step of no-capture, we obtain five profiles of distance evolutions shown in Fig. 8.
Y. Rollot et al. / International Journal of Adhesion & Adhesives 19 (1999) 35—48
41
Fig. 8. Profiles of distances evolution.
When considering Van der Waals and pull-off forces, the fives characteristic stages can be drawn out as follows: (i) first capture F 'F , \ (ii) slight detachment between the ball and the probe with D (D : capture (3 (3 D= "½= ! D= '0 N ½= ' D= (0) 2 2
(iii) large detachment between the ball and the probe with D 'D : capture D= 'D= N ½= '(3D= (0) (iv) no capture: D is always greater than D , (v) no capture and no detachment between balls:
Stages (i) and (iv) can be considered as critical points corresponding to limit external forces applied to the probe. The lower limit comes from inequation (16), which, applied to Eq. (1) gives
(3
!
2
D= (1)*t!2
(3 R\ "D (0)#½= *t! D= (i)*t 2 G
(18)
In the same way, for D (t): R\ D (t)"D (0)# D= (i) *t G
(19)
k º" ( L (D #½= *t! L\ º *t) G G k ! (D # L\ º *t) G G
(20)
The series S associated to º is defined by S " L\ º . L L L G G Expression (20) then becomes
A R m ½= # #m g5F 6D
k º" L ( (D #½= *t! S *t) L\
1 A R N ½= 5 F - ! m 6D
where D "4;10\ m and ½Q "½= *t. The upper limit depends on more complex relationships. To evaluate this limit, Eq. (2) is written as follows: k k D= (t)" ! D (t) D (t)
(3 D (t)"D (0)# ½= ! D= (0) *t 2
Replacing expressions (18) and (19) in Eq. (17), a series is obtained:
D= (1)"!D= (0)
equation:
(17)
where k "2.67;10\, k "1.78;10\, D= (0) "5.59;10 m s\ and D (0)"D (0)"D . Considering the speed profile of the probe to be a step, D at instant t can be expressed by the following
k ! (D #S *t) L\
(21)
The aim is to define the upper limit of the external force ensuring the capture. This is obtained when D (t) and D= decrease, i.e.: º 'º ∀n. L L> This series cannot be written as a recurring series. The problem can be summarized in three points: z ½= is the unknown of the problem, z º is a descending series, L z S is a growing series, then º (S ((n!1)º L L\
42
Y. Rollot et al. / International Journal of Adhesion & Adhesives 19 (1999) 35—48
using Eq. (21) k º' L ( (D #½= *t! S *t!º *t) L\ L k ! (22) (D #S *t#º *t) L\ L The resolution of this last equation with MapleV gives two results but only one compatible with the problem. The corresponding solution is ½$ *t'0.92;10\#0.22;10\S L\ The minimum value is then obtained by reducing S and replacing it by º . The upper speed limit L\ ensuring the capture is then ½Q "9.31;10\ m s\ By simulation, this speed limit has been found to be » +9.34;10\ m s\.
4.2. The capture task (VdW, capillary, electrostatic and pull-off forces) For this model, contact electrification and capillary phenomena are added. Humidity is fixed at 50%
(r "1.6 nm). The calculated values of pull-off forces
are: F "6.67;10\ N and \ F "3.72;10\ N \ The lowering of the pull-off force magnitudes (compared to previous model) is due to the condensed water on surfaces which decreases adhesion work. These pull-off forces are weak compared to the capillary force magnitudes. In this case, the external force applied to the probe is always greater than the pull-off forces (Eq. (1) gives the minimum value for the external force equal to 5.2;10\N). Ball 1 is then captured at a weak value of ½Q . There is no minimum critical time law for the capture. Experimentally, the detachment kinetic may increase the magnitude of the pull-off forces and thus a lower critical speed limit may arise reducing the admissible speed range. When the step value is increased, a value at which no capture occurs is reached. By simulation, this limit is found to be between the values 2.3;10\ m s\ and 2.4;10\ m s\ (Fig. 9). Figs. 10 and 11 show the evolution of VdW, capillary and electrostatic forces vs time for both interactions, ball—ball and ball—probe and the twostep value limits.
Fig. 9. Evolution of distances for the transition between capture (2.3;10\ m s\) and no-capture (2.4;10\ m s\).
Fig. 10. Evolution of ball—ball forces for step values 2.3;10\ and 2.4;10\ m s\.
Y. Rollot et al. / International Journal of Adhesion & Adhesives 19 (1999) 35—48
43
Fig. 11. Evolution of ball—probe forces for step values 2.3;10\ and 2.4;10\ m s\.
For the step value 2.3;10\ m s\, the appearance of the ball-probe electrostatic force between instants 20 and 30 ns causes the decreasing of D= . This is after all ob served in the D profile. The second derivative of geomet ric relation (4) forces D= to evolve in the opposite way. At time 60 ns, D becomes greater than the electron tunnel ling distance (D ) and the ball—ball electrostatic force arises (Fig. 10). This force, whose magnitude decreases rapidly is not strong enough to oppose ball—probe forces. As D is invariant between 60 and 120 ns, it allows D to linearly increase. Ball—ball forces are then too weak to oppose ball—probe forces and the capture occurs. For the step of value 2.4;10\ m s\, the probe-ball electrostatic force which appears (Fig. 11), increases D= . This phenomenon is observed on the D profile between instants 40 and 60 ns. D= decreases under the influence of the ball—ball electrostatic force. This decreasing, via Eq. (4), allows D to increase and the magnitudes of ball—probe forces to decrease rapidly. Ball—ball forces dominate the equilibrium of ball 1, bringing it into contact with balls 2 and 3. Capture does not occur.
If the step value is increased, an irreversible break in the probe—ball adhesion is reached. By simulation this limit has been found in the range 6.9;10\ —7;10\ m s\. This is smaller than the one found for the ‘‘VdW—pull-off’’ model. In the first instants of the manipulation, electrostatic forces are non-existent and the model is the same as the ‘‘VdW—pull-off’’ model except Eq. (11) related to D= . Rubbing forces decrease D= and because of the second derivative of Eq. (4) they increase D= . The gap between D and D increases with the magnitude of the applied step till the adhesion breaks definitively at 7;10\ m s\. Deformations are not negligible compared to the critical distance D (d +3.8 nm, d + \ \ 1 nm). Electrostatic forces then appear too late to oppose the increase of the gap and to permit the manipulation by adhesion.
4.3 The capture task (VdW, electrostatic, friction and pull-off forces)
As it has been shown previously, the probe’s speed plays an important parameter in the object capture. The following is consecrated to the influence of this speed on the release task. In order to achieve the release task, the probe is tilted at an angle h. ½Q and h are the problem’s variables. The task is realised in a humid environment. Thus, Van der Waals, capillary and electrostatic forces are taken into account. Manipulation time is short enough to ignore ball—substrate deformation. Nevertheless, the probe—ball deformation imposes a pull-off condition expressed by
Capillary forces do not appear in this model. The behaviour of the admissible manipulation speed thus follows the ‘‘Vdw—pull-off’’ model. This indicates that each force has comparable magnitude and confirms capillary forces predominance compared to electrostatic and VdW forces. The rubbing forces are only present on contact. Their influence is then limited to the primary instant of the manipulation. When contact is broken only VdW and electrostatic forces act. Rubbing coefficient equals 0.2. For this value, the first step of capture equals the one of the ‘‘VdW—pull-off’’ model: 5;10\ m s\. This step value changes if the rubbing coefficient is increased to 0.9. For lower values, rubbing does not play any role in the magnitude of the external force needed for the first capture.
4.4. The release task
F 'F cos(h) \ -
(23)
The probe is in gold (soft material) thus deformations are important. Calculation of d gives d "37.75;10\ m
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Y. Rollot et al. / International Journal of Adhesion & Adhesives 19 (1999) 35—48
Gold and Silicon possess a high Hamaker constant. Despite water in the interface, the pull-off force is important: F "9.3;10\ N. \ The release is efficient if D "D (ball—substrate dis tance) and if D '0 (probe—ball distance). The curves in this part will show the evolution of D , D and d . This last value will represent the evolution of the central displacement vs time. Distance D will equal zero if d'0. Simulations have been carried out. They show that because of capillary forces, very weak step values allow the release. Thus, a step of value 10\ m s\ combined with an angle h3[88°, 90°] makes the release efficient. As can be predicted with Eq. (23) the angle range increases with the value of the step applied. For example, this band is [76°, 90°] for a step of 10\ m s\. The specificity of these low speeds can be seen on the D profile. No bump is observed at any angle of the band imposed to the probe. The release speed limits vs angle h can be calculated with Eq. (23): m ½= k k k # # # cos h D 2r #D D
m g # 'F \ cos h
4.5. Conclusions on simulations
This inequation has to be verified when D "D . At this distance, the weight is negligible compared to other forces, and k , which represents the electrostatic force equals zero. Constants of the previous equation have then the following values: k "1.69;10\, k "1.47;10\,
the release. When the step value is increased, bumps on the D profile occur, indicating a breaking in ball—substrate contact. The magnitude of this detachment decreases when h increases but is always present for speeds greater than 10\ m s\. Distance characteristic profile evolutions vs. time are shown on the following curves for high applied speed. The applied speed is ½Q "10\ m s\ and imposed angles h"65°, 68° and 70°. Fig. 12b illustrates a case where the release does not occur. Figs. 13 and 14 show two cases of release. The relationship which exists between displacement speed and probe angle, in the aim of calculating the upper speed limit of release, is complicated. It is related to the discontinuity of electrostatic forces and accelerations ½= , D= and D= . Reasoning on the same basis as in the ‘‘VdW—pull-off’’ capture model could nevertheless be carried out. To ensure a precise release, it is important to avoid a bump on the D profile. For low displacement speeds (lower than 10\ m s\), breaking in the ball—substrate contact can be avoided by imposing a high value of h (greater than 80°). Choosing a low displacement speed and a high angle h guarantees a precise release.
k "0
Results are collected on the curve (Fig. 12a) that shows the minimum speed vs angle applied to guarantee
As it was explained in previous sections, the motion’s kinetics may act on the pull-off values and then change the speed limit values. These limits must be confirmed by a set of rigorous experiments. The model proposed here can be generalized for other manipulations and materials. Fig. 15 summarises the speed range limits applied to the system which ensure the capture by adhesion when different forces are considered. More details on the release are collected in Fig. 12a.
Fig. 12. (a) Lower limit for the release task as a function of h and speed, (b) Evolution of distances for a step value 10\ m s\ and h"65°.
Y. Rollot et al. / International Journal of Adhesion & Adhesives 19 (1999) 35—48
45
Fig. 13. Evolution of distances for a step value 10\ m s\ and h"68°.
Fig. 14. Evolution of distances for a step value 10\ m s\ and h"70°.
Fig. 15. Summary of speed range ensuring capture by means of different force combinations.
5. Parameters sensitivity
5.1 Influence of materials combination
In this section, parameters of the system are more precisely studied. The studies have been carried out in the most simple case with the ‘‘VdW—pull-off’’ model. These studies pointed out useful information on the manipulation by adhesion.
Two inequations on Hamaker constants facilitating manipulation by adhesion have been extracted from a static consideration: A 'A
and
A '(3A /2
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Y. Rollot et al. / International Journal of Adhesion & Adhesives 19 (1999) 35—48
In previous sections, materials were chosen to respect these inequations. In the aim to observe the role of the interfaces, constitutive materials of different parts of the system have been changed. First, the following combination was chosen: z probe in polystyrene, balls in silicon, substrate in gold. This choice allowed no manipulation. Accelerations D= and D= are negative (Eqs. (2) and (3)), and distances D and D always equal the contact value D . In this case, adhesion probe-ball is too weak to oppose ball—ball and ball—substrate adhesions. This example indicates that when manipulating in a dry environment, a strong adhesion on the probe is needed. This implies that the probe must be made of a high Hamaker constant material. This conclusion is also valid in a humid environment. Indeed, for the two configurations of capture: z ¹he capture of ball 1: In this case, if D " \ D "D then F 'F eases the cap \ \ \ ture task. z ¹he capture of balls 2 and 3: In this case, if D \ "D "D then F "F \ \ \ keeps the predominence of F 45 in the capture \ task. A second combination has been simulated: z Probe in gold, balls in polystyrene, substrate in silicon. This choice guarantees a strong adhesion for the ball—probe interface and decreases the ball—substrate adhesion. The general behaviour during simulation is close to the one obtained with the gold—silicon—polystyrene combination. In fact, distance D is constant at the contact value D . This behaviour comes from Eq. (3). At contact, the weight is negligible. The equation is positive if: (3A
R A R ' (24) 24D 6D Hamaker’s combination relationships are supposed to be applicable. A can then be related to A and A : A "(A A 1 Putting in inequation (24): (3A
/4'(A A thus A '16A /3 With this condition, distance D increases, indicating that balls contacting the substrate are also captured. This hypothesis is not suitable for the task to be achieved. Simulation results then show comparable behaviour but characteristic values are shifted forward. The first capture occurs for a step value of 6;10\ m s\, and the last capture occurs for a step value of 2.1;10\ m s\. This shift is due to the decreasing of the ball—ball pull-off
force (1.93;10\ N vs 7.68;10\ N). The capture band is increased because of the lower limit variation. This points out the importance of the pull-off force as a condition for the manipulation. Computing the magnitude of this force depends on Hamaker’s constant and adhesion energies. These characterize the range of admissible manipulation speeds. Nevertheless, in a humid environment, the lower limit does not exist (in a first approximation, neglecting kinetic effects during detachment at the microscale). Variation on the band by changing the material combination is then very weak. The necessary condition ensuring manipulation by adhesion is the choice of a probe with a high Hamaker’s constant compared to the one of the balls and the substrate. 5.2 Influence of sphere radius This section is consecrated to the study of the influence of the sphere radius on the manipulation task. This study is always carried out with the ‘‘VdW—pull-off’’ model. Eqs. (1)—(4) can be written as follows: m ½= "F !C R !m g
(25)
(3 C C D= # D= " !C R 2
(26)
C C D= " !C R
(27)
(3 ½ "D #2R # (D #2R )#D 2
(28)
Influence of the sphere radius on the minimum speed of capture is obtained by applying the pull-off condition (16) as follows: R ½= 5 (n¼ !C ) where m \ C "A /(6D), D "4;10\ m and ½Q "½= *t This expression shows that the minimum acceleration (i.e. the minimum speed) is linearly related to R . The upper limit is extracted from the following equation:
1 k k ! D= (t)" R D (t) D (t)
(29)
where R"R /5;10\ is the radii ratio (the initial radius is R "5 lm), k and k are the variables expressed in the section concerning the simulation of the ‘‘VdW—pull-off’’ model, D (0)"D (0)"D and D= (0)"(1/R);5.594;10 m s\.
Y. Rollot et al. / International Journal of Adhesion & Adhesives 19 (1999) 35—48
47
Fig. 16. Speed limits ensuring capture vs sphere radius for the ‘‘VdW—pull-off ’’ model.
D at time t can be written as follows if the speed profile is a step: (3 R\ D (t)"D (0)#½= *t! D= (i)*t 2 G In the same way for D (t): R\ D (t)"D (0)# D= (i)*t G Replacing these two expressions in Eq. (29), a series is obtained:
k 1 º" L R (D #½= *t!(S *t) L\ k ! (30) (D #S *t) L\ The aim is to define the limit of the capture force. D (t) and D= must decrease, thus: º 'º ∀n L L> After simplifications, this equation is no more related to R. Solution obtained in section ‘‘dynamic models of the capture task considering only Van der Waals forces’’ is then still valid and can be expressed:
½= *t'0.919;10\#0.2206;10\S (31) L\ where S can be reduced to D= /R. L\ The growing of radius brings to a decreasing of the maximum admissible speed of capture. The speed of
capture range shrinks till it becomes nil. Evolutions obtained by simulation are summarized in Fig. 16. The maximum radius of a silicon sphere allowing the manipulation mode proposed when only VdW and pulloff forces are taken into account is then R +120 lm. Conclusion Within the framework of manipulation of micro-objects, a manipulation mode was established. It has been theoretically validated by simulations using a general dynamic model of micro-manipulation. Some strong conclusions have been extracted on material combinations, geometry and speed for the feasibility and existing conditions. The next step of this work will be to carry out, in a free environment, experiments associated to the example developed in this paper. Reflection on possible components of the system, and particularly on the part capable of generating the accelerations needed for the capture task have permitted a final glimpse of the system. Other applications of this system are foreseen, for example, a tool which determines the Hamaker constants of different materials, or the kinetic behaviour of the pull-off forces at the micro-scale.
Acknowledgements To M. Barquins, Research Director at the CNRS, for the fruitful discussion on the topic and to Sue McCarthy for her help.
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References [1] Rollot Y, Regnier S, Guinot J-C. Microrobotique: mode`le dynamique et loi horaire pour une micromanipulation par adhe´sion. C. R. Acad Sci Paris Se´r II 326 b, 1998;469—74. [2] Hamaker HC. The london Van der Waals attraction between spherical particles. Physica 1937;10:1058—72. [3] Israelachvili JN. The nature of Van der Waals forces. Contemp Phys 1974;15(2):159—77. [4] Adams MJ, Perchard V. The cohesive forces between particles with interstitial liquid. Int Chem E. Symp Ser No. 91, 1963; 147—160. [5] Mastrangelo C. Mechanical stability and adhesion of microstructures under capillary forces — Part I: Basic theory. J Microelectromech Systems 1993;2(1):33—43. [6] Lowell, J, Akande A. Contact electrification — why it is variable?. J Phys D 1988;21:125—37. [7] Horn RG, Israelachvili JN, Pribac F. Measurement of the deformation and adhesion of solids in contact. J Colloid Interface Sci 1987;115(2):480—92. [8] Maugis D. Adhesion of spheres: the JKR-DMT transition using a dugdale model. J Colloid Interface Sci 1992;150(1): 243—69.
[9] Derjaguin BV, Muller VM, Toporov YU P. Effect of contact deformations on the adhesion of particles. J Colloid Interface Sci 1975;53(2):314—26. [10] Ando Y, Ishikawa Y, Kitahara T. Friction characteristics and adhesion force under low normal load. J Tribol 1995;117:569—74. [11] Feynman RP. There’s plenty of room at the bottom. Talk at the Annual Meeting of the American Physical Society at the California Institute of Technology (Caltech), December 29th, 1959. [12] Kaeble DH. Physical chemistry of adhesion. New York; Wiley Interscience Publication, 1971. [13] Israelachvili J. Intermolecular and surface forces. New York: Academic Press, 1991. [14] Lowell J, Rose-Innes AC. Contact electrification. Adv Phys 1980;29(6):947—1023. [15] Hays DA. Electrostatic adhesion of non-uniformly charged dielectric sphere. Int Phys Conf Ser. No. 118: Sec. 4, 1991: 223—8. [16] Lowell J, Akande A. Contact electrification of metals. J Phys D 1975;8:53—63. [17] Krupp H. Particle adhesion: theory and experiments. J Adv Colloid 1967;1:111—239. [18] Johnson KL, Kendall K, Roberts AD. Surface energy and the contact of elastic solids. Proc R Soc Lond 1971;A 324:301—13.
Appendix Numerical values for different Where: o Volumic weight (10 kg/m) e dielectric constant E electronic affinity (eV) c surface energy (mJ/m)
components K stiffness (;10 Pa) n refraction index E Ionization potential (eV) ' A Hamaker’s constant (10\ J)
M molar weight a polarizability (10\ cm)
+(E #E )/2 work function (eV) ' k viscosity
Materials
o
e
n
a
E '
E
c
A
K
M
Gold (Au) Polystyrene (PS) Silicon (Si)
19.283 1.05 2.33
— 2.55 11.7
— 1.55 3.41
5.8 15 5.38
5.1 4.9 4.85
9.23 8.43 8.15
2.31 1.37 1.385
1450 35.5 1400
54.6 7.29 25.8
2.5 80 140
196.97 104.15 28.0855
Water o
e
n
E '
a
M
k
0.9975
80
1.333
12.612
1.45
18.05
1.854
Hamaker constant’s for different contacts A 1
A
33.6
13.2
1 .1
A 5 1
A
20.3
13.8
1 5 1
A 1 5 .1 0.27