Capillary Forces in Microassembly: Modeling, Simulation, Experiments, and Case Study

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microtechnology and mems

microtechnology and mems Series Editor: H. Fujita D. Liepmann The series Microtechnology and MEMS comprises text books, monographs, and state-of-the-art reports in the very active ﬁeld of microsystems and microtechnology. Written by leading physicists and engineers, the books describe the basic science, device design, and applications. They will appeal to researchers, engineers, and advanced students. Mechanical Microsensors By M. Elwenspoek and R. Wiegerink CMOS Cantilever Sensor Systems Atomic Force Microscopy and Gas Sensing Applications By D. Lange, O. Brand, and H. Baltes Modelling of Microfabrication Systems By R. Nassar and W. Dai Micromachines as Tools for Nanotechnology Editor: H. Fujita Laser Diode Microsystems By H. Zappe Silicon Microchannel Heat Sinks Theories and Phenomena By L. Zhang, K.E. Goodson, and T.W. Kenny Shape Memory Microactuators By M. Kohl Force Sensors for Microelectronic Packaging Applications By J. Schwizer, M. Mayer and O. Brand Integrated Chemical Microsensor Systems in CMOS Technology By A. Hierlemann

CCD Image Sensors in Deep-Ultraviolet Degradation Behavior and Damage Mechanisms By F.M. Li and A. Nathan Micromechanical Photonics By H. Ukita Fast Simulation of Electro-Thermal MEMS Efﬁcient Dynamic Compact Models By T. Bechtold, E.B. Rudnyi, and J.G. Korvink Piezoelectric Multilayer Beam-Bending Actuators Static and Dynamic Behavior and Aspects of Sensor Integration By R. Ballas CMOS Hotplate Chemical Microsensors By M. Graf, D. Barrettino, A. Hierlemann, and H.P. Baltes Capillary Forces in Microassembly Modeling, Simulation, Experiments, and Case Study By P. Lambert

P. Lambert

Capillary Forces in Microassembly Modeling, Simulation, Experiments, and Case Study

123

Professor Pierre Lambert Université Libre de Bruxelles (ULB) BEAMS Department (CP165/14) Avenue F.D. Roosevelt 50 1050 Bruxelles, Belgium Series Editors: Professor Dr. Hiroyuki Fujita University of Tokyo Institute of Industrial Science 4-6-1 Komaba, Meguro-ku Tokyo 153-8505, Japan Professor Dr. Dorian Liepmann University of California Department of Bioengineering 6117 Echteverry Hall Berkeley, CA 94720-1740, USA

Library of Congress Control Number: 2007927260 ISBN 978-0-387-71088-4

e-ISBN 978-0-387-71089-1

Printed on acid-free paper. © 2007 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 9 8 7 6 5 4 3 2 1 springer.com

I dedicate this book to those whose time I devoted to writing it. Je d´edie ce livre `a ceux a` qui j’ai pris le temps de l’´ecrire.

Foreword

Within the ﬁeld of microassembly, this book crosses a bridge between the world of surface science and chemistry on the one hand and the world of mechanical engineering on the other hand. Indeed, the mechanical devices produced at a scale ranging from a few micrometer up to a few millimeter are brought face to face with the eﬀects of downscaling, and in particular with the predominance of surface tension eﬀects over the gravity eﬀects. Many illustrations of this trend can be found in the literature and in emerging industrial products based on surface tension eﬀects such as the ﬂuid lens patented by B. Berge and produced by Philips, the emergence of capillary stop drives or, with other words, surface tension based micro-valves, the use of surface tension combined with electrostatic eﬀects in the manufacturing of liquid handling systems such as the EWOD (i.e., electro-wetting on dielectric) devices, and so on. To focus on microassembly, two approaches are currently considered. The self-assembly paradigm, in which surface eﬀects are used to organize and assemble micrometric structures (mainly up to a few micrometer), and the microrobotic assembly, based on the miniaturization of the actuation, high resolution micromanipulators, and gripping devices, more dedicated to mesoscopic sized components (mainly down to about 10 µm). Self-assembly is clearly not the subject of this book, even if some obvious links relate the proposed models to this ﬁeld. As a scientiﬁc knowledge, microrobotics focuses on active structures, able to produce motions and to interact mechanically, i.e., produce eﬀorts, with their environment at the microscale (between a few micrometer and a few millimeter). One of the main challenging issues of it concerns the handling of small components, in order to precisely position, assemble, characterize, or modify them. The research in this ﬁeld covers a wide area of interesting topics, including the exploration of new phenomena (i.e., which are new from the point of view of microrobotics) and the development of an adequate scientiﬁc background (step 1), the development of demonstrators illustrating new strategies to pick up, to handle, and to release microcomponents, and which

VIII

Foreword

try to minimize or take beneﬁt from the new physical eﬀects of the miniaturization (step 2), and ﬁnally, the set up of eﬃcient and reliable industrial products addressing speciﬁc needs (step 3). Step 1 is fundamental in that sense that new eﬃcient micromanipulation systems can only be developed bearing in mind the speciﬁcities of the micro-world and take advantage of it through new approaches. Precisely, this book proposes a physical understanding of the surface tension phenomena, builds models that can be used in simulations and in the design of a surface tension based gripping demonstrator. The author uses wellknown concepts from surface science (like surface tension, capillary eﬀects, wettability, contact angles) and eﬃciently uses them as outputs of chemists models (which explain whether a liquid will wet or not a surface), but as inputs of mechanical models predicting the amount of eﬀort that can be used to handle microcomponents. The book is unique in that sense that this is the ﬁrst in this direction and it proves that the microrobotic approach can lead to very eﬃcient systems. It is very well organized and the content is presented in a very rigorous, pleasant, and pedagogical manner by a real expert of the addressed issues. We strongly recommend to all persons, students, engineers, researchers who are interested in micromanipulation and microassembly to read it.

Besan¸con Brussels Lausanne April 2007

Prof. N. Chaillet Prof. A. Delchambre Prof. J. Jacot

Preface

0.1 Context In the current context of trend to miniaturization, the main goal deﬁned at the very beginning of this work was to study the inﬂuence of miniaturization on the manipulation tasks performed in microassembly, because for a few years, most papers dealing with microassembly have referred to overviews that mentioned the importance of forces related to the microworld. The reader can have a quick overview on the scales covered by the term microworld in Fig. 0.1. In this ﬁgure, several domains can be distinguished: 1. The “macro” domain, related to conventional manufacturing and assembly technologies 2. The “micro” domain where the limits of conventional means can be undergone and new strategies arise. Sometimes the upper area of the micro domain is called “meso” domain 3. The “nano” domain ﬁlls the gap between the micro domain and the atoms and molecules world. It is the ultimate domain of mechanical engineers As a comparison, the accuracy of conventional manufacturing is about 10 µm and the size of hair is between 10 and 100 µm. This book deals with components

meso nano

micro

macro

L(m) 10-9

10-6 µ-accuracy

Fig. 0.1. Sizes and scales

10-3 µ-components

1

X

Preface

ranging from 10 µm to a few millimeter, with part features that can reach the micron: The chosen case study consists in a watch ball bearing with 0.3 and 0.5 mm diameter balls. More generally, the current breakthrough of the miniaturization of electronic components and the development of their related production equipment make it possible today to produce cheap components integrating a lot of functionalities. These production techniques allow the 2D manufacturing to use several materials: glass, silicon, metals. Beside these applications from the semiconductor industry, the conventional mechanical design also tries to reduce the size of the products and the emergence of micromechatronics develops new miniaturized robots with a lot of functionalities (sensing, actuation, guiding). This trend does not spare assembly and the products are not only reduced in size but also the assembly and production equipment are downscaled, giving rise to several concepts like microfactory or new assembly strategies such as parallel assembly. The pieces of equipment and especially the grippers are downscaled, but new grippers based on microworld related physics are now commercially oﬀered by a lot of industries and laboratories. The ﬁrst representation that crosses the mind when talking about micro is that it surely must be “small.” The preﬁx micro can of course be understood as deﬁning the size of a component (10−6 m), but a microproduct has not to be understood as a product with a size of a few microns. Let us give an overview of some deﬁnitions that can help us better deﬁne the concepts of micropart, microcomponent, microproduct, microsystem, microassembly. Benmayour [19] proposes a general deﬁnition of a microproduct using an analogy with the term “microscopic” object. In the same way as a microscopic object cannot be seen with bare eye, a microproduct is a product that can neither be manufactured nor assembled with bare hand: The production of a microproduct requires adapted manufacturing and assembly equipment. Unfortunately, this deﬁnition is quite general and some conventional products like cars cannot be considered as microproducts even when assembled with dedicated equipment. Moreover, this deﬁnition can give us an upper boundary but cannot provide any indications about the lower limit of a microproduct. However, it conveys the idea that the size criterion alone cannot be taken into account. We consider in this book microproducts like a watch ball bearing made of microparts or microcomponents (like balls). Roughly speaking, we will consider that microproducts have sizes ranging from a few cm3 to a few dm3 . For example, we use to speak about a micropump for a product that has external dimensions of a cylinder with a 8 cm diameter and 2 cm height. These microproducts are made of several microparts or microcomponents that have a size ranging from 10 µm to a few millimeter, but they can have some features with a size reaching 1 µm. For example, the pumping mechanism of a micropump can be smaller than a cube with 10 mm edge, having at least one dimension smaller than 100 µm. Nelson [130] generally refers to 1 µm– 100 µm as “microscale” and 100 µm to 1 mm as “mesoscale.”

0.1 Context

XI

As far as assembly equipment is concerned, most microfactories are actually desktop factories, that is having external dimensions of 1 m2 ×40 cm. Bohringer et al. [22] locates the ﬁeld of microassembly between conventional assembly, dealing with part dimensions higher than 1 mm and what they call “the emerging ﬁeld of nanoassembly” (with part dimensions ≤1 µm). A microgripper can be a gripper to handle microcomponents, even if the whole gripping mechanism is still quite big compared to the handled part, or it can refer to the terminal tip(s) of the gripper that is(are) in contact with the microcomponent (for example, a particular kind of micromanipulation tool is the Atomic Force Microscope (AFM): This equipment is not designed like a gripper but several laboratories try to use it to push microcomponents. In this case, the AFM tip can be considered as a gripper, made of a cantilever (100 × 10 × 2 µm3 ) with a tip of conical or pyramidal shape of 10 µm height and a tip radius of about 10 nm). Other criteria can be considered to characterize microcomponents, such as, for example, the required tolerances and clearances in order to ensure the function (the pumping mechanism of the micropump cannot show clearances bigger than a few micron in order to guarantee that drug can be transferred from the tank to the patient). A less quantiﬁable way to deﬁne a micropart is to verify whether the models and the techniques used in the macroworld are still valid. For example, macroassembly is clearly based on the mechanical grip force to pick up and the own weight of the component to release, while microassembly has to turn to other techniques due to relative decrease of the gravity force compared to surface forces (see Fig. 0.2). As the main goal of this work is to consider the modeling of the forces acting in the manipulation of a micropart, we consider that the use of these forces make sense in our microcomponents. We prefer to refer to model assumptions and compare the sizes of a part or the roughness of a component with several cut-oﬀ lengths arising from model assumptions. We consequently identify a domain between a “van der Waals” cut-oﬀ length of a few tens of nanometer

Forces exerted on the component [N]

10

10

Classical gripping

Capillary gripping

5

10

Vacuum gripping

A

0

10

C

-5

B

10

-10

10

Weight ~ L3 10

-15

Vacuum force ~ L2 Capillary force ~ L

-20

10

-8

10

-6

10

-4

-2

10 10 Size of the component [m]

Fig. 0.2. Scaling laws and micromanipulation

0

10

XII

Preface

Table 0.1. Comparison between micro and macroproducts Criterion Size

Macroproduct

Microproduct Below 1 mm Below 500 µm Accuracy 0.1–10 µm 5 µm Clearances Very small Complexity Made of several Multifunctional, complex elementary components products, few components Compact design products Maintenance Maintenance and replacement No maintenance, replacement of the defective components of the product in case of failure Heterogeneousness Several parts from diﬀerent technological domains involving new joining techniques

Ref. [166] [41] [166] [91] [166]

[154]

[166]

Table 0.2. Comparison between micro and macroassembly Criterion Automation Batch size Resource consumption Response time

Macroassembly Microassembly Automatic Manual and semiautomatic, to be automated. Single parts, Batches of parts, serial assembly parallel assembly Expected to be lower

Ref. [65, 166], [159, 185]

[6]

Expected to be shorter because of lower inertia

(limit of the nonretardated van der Waals forces, see page 10) and a capillary cut-oﬀ length of a few millimeter (see (8.1)): This domain will be considered as our microworld. To give the reader a broader overview, we summarize some criteria related to micro/macroproducts and to micro/macroassembly (Tables 0.1 and 0.2).

0.2 Contributions of this Book This book falls into ﬁve parts whose main contributions are summarized in Fig. 0.3 (the ﬁfth part containing the appendices is not shown in this ﬁgure). The ﬁrst part introduces the concept of microassembly (Chap. 1), proposes in Chap. 2 a classiﬁcation of the forces acting in microworld (which has been deﬁned in the previous section), and summaries in Chap. 3 the numerous gripping principles proposed in the scientiﬁc literature. This summary (which is essentially a review of the literature) serves as a basis for a gripping principles classiﬁcation from which it turns out that the forces generated by surface tension can suit the microgripping task.

0.2 Contributions of this Book

XIII

Part I: Microassembly Specificities • Different kinds of microassembly • What are the forces in action • What are the possible handling principles -- classification of the handling principles -- proposal: the capillary gripper

Part II: Modeling and simulation of Capillary Forces • Parameters involved in a gripping based on surface tension • Classical methods for capillary forces computing: energy derivation method, geometrical approximations, resolution of the Laplace equation at equilibrium -- Proof of equivalence between the energy derivation and the Laplace equation based methods -- Implementation of a double iterative numerical scheme to compute forces in the axially symmetric case, based on the solving of the Laplace equation -- Determination of the limits of this static simulation -- Determination of approaching contact distance, rupture distance and residual volumes after rupture -- Approximation of cycle times -- Application to the watch ball bearing case study

Part III: Experimental Aspects Testbench: -- Set up of a force measurement testbench (from 10µN to 10mN) -- Set up of a contact angles measurement testbench -- Tested liquid: water, isopropanol and silicone oil, from 0.1µL to 1µL -- Tested materials: steel, silicon, PTFE, zirconium -- Tested geometries: concave and convex cones, spheres, cylinders

Studied parameters and phenomena: -- Inputs: gap, geometries, contact angles, surface tension, dynamic release, volume, relative orientation, evaporation -- Outputs: forces and liquid bridges profiles

Watch ball bearing case study: -- Study of the picking errors and solutions -- Study of the releasing reliability -- Measurement of the picking force and reliability study

Answered questions: -- Advancing vs receding contact angle, tension term vs. Laplace term -- Quantified comparison between picking principles -- Quantified comparison between releasing strategies -- Design rules for a surface tension based gripper

Part IV: Perspectives Modelling and Simulation - Dynamic simulation - Capillary condensation simulation

Design and manufacturing perspectives - Surface tension control (i.e. electrowetting) - Design and manufacturing of a surface tension based gripper prototype for SMD components

Fig. 0.3. Contributions of this book

XIV

Preface

The second part concerns the modeling aspects. Therefore, Chap. 6 presents the underlying parameters (such as surface tension and contact angles) and models (Young-Dupr´e and Laplace equations), which rule the surface tension forces (also called capillary forces). This chapter explains the action of these forces on a solid, thanks to two terms: the so-called “Laplace” or pressure term and the so-called “interfacial tension” term (see Sect. 6.5). Based on these parameters, Chap. 7 reviews some approximations to compute capillary forces at equilibrium: energy diﬀerentiation methods, geometrical methods assuming a given shape of the meniscus (typically arc or parabola). Chapter 8 details how to implement a numerical resolution of the so-called Laplace equation to determine the meniscus shape in axially symmetric cases. This allows the computation of the capillary forces linking a component and a gripper, relying on the following assumptions: equilibrium, vanishing Bond number (i.e., gravity is neglected), axial symmetry, constant contact angles, constant volume of liquid. The originality of this model relies on the fact that the volume of liquid can be imposed, which leads to a double iterative scheme for the resolution. Another contribution of this book is to prove analytically the equivalence of this approach and the energy minimization method (in the case of a prism–plane interaction, see Chap. 9). The proposed model is applied to the case study of a watch ball bearing, showing the interest for a gripper geometry conforming with that of the component (Chap. 10). This model is then enriched, thanks to a second set of parameters (Chap. 11), showing how surface roughness and surface impurities can be included in the model through the value of the contact angle. The contact angle hysteresis is introduced in this chapter; however, it will be shown (thanks to experiment) how to chose between both. Finally, this chapter illustrates with a ﬁgure from the literature an interesting damping eﬀect, which prevents high contact forces. The limits of the proposed model are discussed in Chap. 12, showing the suitability of this model even in the case of highly accelerated components. This chapter provides some approximations of the damping time of the oscillations of the meniscus, which indicates a ﬁrst order of magnitude of the cycle time of a surface tension based picking task. Some conditions of meniscus rupture are given in Chap. 13. To conclude this second part, a detailed implementation of the proposed models is given in Chap. 14. The third part of this book focuses on experimental aspects. First, we detail in Chap. 17 the set up of an experimental test bed allowing the measure of the models inputs (contact angles, volumes of liquid) and outputs (forces and meniscus shapes). Then, Chap. 18 provides numerous model validation and exhaustive results concerning the inﬂuence of the gap, the gripper geometry, the surface tension, the contact angles (including the choice between the advancing and the receding contact angles), the relative orientation of the gripper with respect to the component, the conditions of dynamical release, and the rupture distance of the meniscus. Theses results are discussed in Chap. 21 in terms of picking and releasing strategies; therefore, we introduce the concept of adhesion ratio φ:

0.3 What this Book Does Not Tell

φ=

Fmin , Fmax

XV

(0.1)

where Fmin and Fmax are, respectively, the minimal and the maximal values of the capillary force, which is assumed to be tuned between the picking stage (Fmax ) and the releasing stage (Fmin ). Ratios tending to zero indicate a very ﬂexible gripping strategy (components with a large mass range can be picked), while a ratio tending to 1 indicates a nonsuitable gripping strategy. These results have been then applied in a ﬁnal illustration of the surface tension gripping based on a watch ball bearing case study. The characterization of the underlying parameters is led in Chap. 19 while Chap. 20 presents the results of picking and releasing tasks of the 0.3 and 0.5 mm diameter balls of this bearing. The conclusions presented in Chap. 21 discuss the results of Chaps. 18, 19, and 20. The fourth part contains the general conclusions and the perspectives of this work (Chap. 22). Finally, the ﬁfth part contains the appendices, which includes modeling and geometry complements, some elements of the proof of equivalence of both capillary force models, some tracks toward a dynamical simulation, and ﬁnally, a list of the main symbols and abbreviations used in this book. The book is ended by a list of references and an index.

0.3 What this Book Does Not Tell This book is an attempt to present on a comprehensive way the elements ruling a reliable surface tension based gripping of a small component with a gripper (typically a sub-millimeter sized component), in gaseous environment (typically ambient atmosphere). However, the analysis proposed to understand the role of the underlying parameters ruling capillary forces is very general, and the proposed model is only valid for axially symmetric cases. In a whatever geometrical conﬁguration, the reader will have to turn himself (herself) toward an energy minimization tool such as, for example, the well-known Surface Evolver software. The case of lateral capillary forces is hardly treated in the experimental part, and we refer the interested reader to the work of Peter A. Kralchevsky [105]. On the same way, the so-called self-assembly or auto-assembly is not treated in this book: These aspects of self-assembly, which are not restricted to capillary forces, are presented, for example, in the work of Karl F. B¨ ohringer. It will be shown that a static modeling is quite suﬃcient for our purpose; nevertheless, the reader will ﬁnd additional information concerning dynamical simulation in [156]. Finally, the case of immersed environments is treated in [64]. Let us note that the example treated in this book concerns the case of watch bearing balls with a diameter ranging from 0.3 to 0.5 mm. The use of surface tension has an upper limit (the so-called capillary length equal to a

XVI

Preface

few millimeter for water), it is not limited in terms of miniaturization. Nevertheless, the manufacturing of micron-sized grippers would require adapted manufacturing techniques that have not been considered in this book, but this is more a perspective than a limitation.

0.4 Reading Suggestion For a quick reading, the chapters and sections listed in Table 0.3 are essential for a good understanding of this book. Let us emphasize the presentation of four examples (Table 0.4). Table 0.3. Quick reading suggestions Chapter/Section Title Page Preface 3 Handling Principles for Microassembly 13 6 First Set of Parameters 41 7.1 Introduction to the State of the Art 51 on the Capillary Forces Models 8 Static Simulation at Constant Volume of Liquid 65 17 Test bed and Characterization 143 21 Final discussion of Part III 211 22 Conclusions and Perspectives 221 Appendix D List of symbols 247

Table 0.4. Examples Chapter 10 14 19 20

Title Page Application to the Modeling of Microgripper for Watch Bearings 83 Numerical Implementation of the Proposed Models 127 Watch Bearing Case Study: Characterization 189 Watch Bearing Case Study: Results 199

Brussels April 2007

P. Lambert

Contents

Preface 0.1 0.2 0.3 0.4

..................................................... Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contributions of this Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What this Book Does Not Tell . . . . . . . . . . . . . . . . . . . . . . . . . Reading Suggestion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IX IX XII XV XVI

Part I Microassembly Speciﬁcities 1

From Conventional Assembly to Microassembly . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Design of Monolithic Products for Microassembly . . . . . . . . . 1.3 Combined Part Manufacturing and Assembly . . . . . . . . . . . . 1.4 Product External Assembly Functions . . . . . . . . . . . . . . . . . . . 1.5 Product Internal Assembly Functions . . . . . . . . . . . . . . . . . . . 1.6 Stochastic or Self-Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Parallel Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 4 6 6 6 7 8 8

2

Classiﬁcation of Forces Acting in the Microworld . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Classiﬁcation Schemes of the Forces . . . . . . . . . . . . . . . . . . . . . 2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 10 12

3

Handling Principles for Microassembly . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Presentation of Gripping Principles . . . . . . . . . . . . . . . . . . . . . 3.3 Classiﬁcation of Gripping Principles . . . . . . . . . . . . . . . . . . . . 3.4 Comparison between Gripping Principles . . . . . . . . . . . . . . . . 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 13 25 28 29

4

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

XVIII Contents

Part II Modeling and Simulation of Capillary Forces 5

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

6

First Set of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Young–Dupr´e Equation and Static Contact Angle . . . . . . . . 6.4 Laplace Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Eﬀects of a Liquid Bridge on the Adhesion Between Two Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 A Priori Justiﬁcation of a Capillary Gripper . . . . . . . . . . . . . 6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 41 42 43

7

45 47 49

State of the Art on the Capillary Force Models at Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Energetic Approach: Interaction Between Two Parallel Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Energetic Approach: Other Conﬁgurations . . . . . . . . . . . . . . . 7.4 Geometrical Approach: Circle Approximation . . . . . . . . . . . . 7.5 Geometrical Approach: Parabolic Approximation . . . . . . . . . 7.6 Comparisons and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 55 57 61 61

8

Static Simulation at Constant Volume of Liquid . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Description of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Equations and Numerical Simulation . . . . . . . . . . . . . . . . . . . . 8.5 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 65 66 67 71

9

Comparisons Between the Capillary Force Models . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Qualitative Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Analytical Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Deﬁnition of the Case Study . . . . . . . . . . . . . . . . . . . . . 9.3.2 Preliminary Computations . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Determination of the Immersion Height h . . . . . . . . . 9.3.4 Laplace Equation Based Formulation of the Capillary Force . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.5 Energetic Formulation of the Capillary Force . . . . . . . 9.3.6 Equivalence of Both Formulations . . . . . . . . . . . . . . . . 9.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 73 73 75 75 76 77

51 51

79 79 80 81

Contents

10 Example 1: Application to the Modeling of a Microgripper for Watch Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Presentation of the Case Study . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Analytical Model Based on the Circle Approximation . . . . . 10.4 Numerical Model Based on the Laplace Equation . . . . . . . . . 10.5 Benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Pressure Diﬀerence Saturation . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XIX

83 83 83 86 89 93 94 96

11 Second Set of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Surface Heterogeneities and Surface Impurities . . . . . . . . . . . 11.3 Surface Roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Static Contact Angle Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Dynamic Spreading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97 97 97 98 99 100 101

12 Limits of the Static Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Performances of the Assembly Machines . . . . . . . . . . . . . . . . . 12.3 Nondimensional Numbers and Buckingham π Theorem . . . . 12.4 Another Approach: Use of a 1D Analytical Model . . . . . . . . 12.5 Limitations of the Static Model . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103 103 103 103 106 108 110

13 Approaching and Rupture Distances . . . . . . . . . . . . . . . . . . . . 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Approaching Contact Distance . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Rupture Distance and Residual Volume of Liquid . . . . . . . . . 13.4 Mathematical and Notation Preliminaries . . . . . . . . . . . . . . . . 13.5 Volume Repartition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6 Rupture Condition and Rupture Gap . . . . . . . . . . . . . . . . . . . 13.7 Analytical Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.8 Summary of the Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.9 Comparison between the Methods . . . . . . . . . . . . . . . . . . . . . . 13.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 111 111 113 114 115 117 119 120 122 124

14 Example 2: Numerical Implementation of the Proposed Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Liquid Bridge Simulation for the Analysis of a Meniscus . . . 14.3 Evaluation of the Double Iterative Scheme . . . . . . . . . . . . . . . 14.4 Pseudodynamic Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127 127 127 131 133 135

XX

Contents

15 Conclusions of the Theoretical Study of Capillary Forces

137

Part III Experimental Aspects 16 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141

17 Test Bed and Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3 Test Bed Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.1 Force Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.2 Drop Dispensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.3 Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4 CAD Model and Drawings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5 Characteristics of the Force Measurement Set Up . . . . . . . . . 17.5.1 Typical Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5.2 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5.3 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5.4 Inﬂuence of a Misalignment on the Force Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.6 Characteristics of the Contact Angles Measurements . . . . . . 17.7 Surface Tension Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 17.8 Modus Operandi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.9 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.9.1 Set of Available Grippers . . . . . . . . . . . . . . . . . . . . . . . . 17.9.2 Set of Available Components . . . . . . . . . . . . . . . . . . . . 17.9.3 Set of Available Blades . . . . . . . . . . . . . . . . . . . . . . . . . . 17.9.4 Available Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.9.5 Contact Angles Characterization . . . . . . . . . . . . . . . . . 17.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143 143 143 145 145 146 148 148 151 151 151 152 152 154 155 155 158 158 159 160 161 161 162

18 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Preliminary Results: Validation of the Simulation Code . . . 18.2.1 Meniscus Proﬁle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.2 Comparison with the Analytical Expressions . . . . . . . 18.2.3 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Advancing vs Receding Contact Angle . . . . . . . . . . . . . . . . . . 18.4 Inﬂuence of the Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4.1 Force–Distance Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4.2 Tension Force vs. Laplace Force . . . . . . . . . . . . . . . . . . 18.5 Inﬂuence of the Gripper Geometry . . . . . . . . . . . . . . . . . . . . . . 18.6 Inﬂuence of the Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . 18.7 Inﬂuence of the Contact Angle θ1 . . . . . . . . . . . . . . . . . . . . . . .

163 163 163 163 164 166 168 170 170 171 171 172 174

Contents

XXI

18.8 Inﬂuence of the Relative Orientation . . . . . . . . . . . . . . . . . . . . 18.9 Auxiliary PTFE Tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.10 Dynamical Release . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.10.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.10.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . 18.11 Approaching Contact and Rupture Distances . . . . . . . . . . . . 18.12 Shear Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.13 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

174 176 177 177 182 185 186 187

19 Example 3: Application to the Watch Bearing Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Available Grippers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3 Available Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4 Liquid Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.5 Liquid Dispensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.6 Contact Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

189 189 189 191 191 192 195

20 Example 4: Application to the Watch Bearing Case Study: Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2 Picking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.2 Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.4 Automated Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3 Placing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4 Compliance Eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.5 Force Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.5.2 Modiﬁcation of the Force Measurement Test Bed . . . 20.5.3 Comparison Between Models and Experiments . . . . . 20.5.4 Ongoing Experimental Study . . . . . . . . . . . . . . . . . . . . 20.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

199 199 199 199 200 201 202 204 205 206 206 206 206 208 209

21 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Picking Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Releasing Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Design Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

211 211 211 213 215

XXII

Contents

Part IV General Conclusions and Perspectives 22 Conclusions and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

221 221 223

Part V Appendices A

Modeling Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Analytical Approximations of the Capillary Forces . . . . . . . . A.1.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.2 Between a Sphere and a Plane . . . . . . . . . . . . . . . . . . . A.1.3 Between Two Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Volume Repartition by the Energetic Approach . . . . . . . . . . A.2.1 Assumptions, Notations, and Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.2 L–V Interfacial Energy . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.3 Total Interfacial Energy . . . . . . . . . . . . . . . . . . . . . . . . .

227 227 227 228 230 233

B

Geometry Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 Area and Volume of a Spherical Cap . . . . . . . . . . . . . . . . . . . . B.2 Diﬀerential Geometry of Surfaces . . . . . . . . . . . . . . . . . . . . . . . B.2.1 Mean Curvature of a Surface . . . . . . . . . . . . . . . . . . . . . B.2.2 Mean Curvature of an Axially Symmetric Surface . . B.3 Catenary Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

237 237 238 238 239 240

C

Comparison Between Both Approaches . . . . . . . . . . . . . . . . .

243

D

Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

247

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

251

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

261

233 234 235

1 From Conventional Assembly to Microassembly

1.1 Introduction The goal of this chapter is to give an overview of diﬀerent assembly strategies that can be used at the considered scale from 10 µm to 10 mm. Indeed, even in the ﬁeld of microproducts, components have to be assembled. The production of microsystems integrating many functionalities, many components made of diﬀerent materials require ﬂexible, modular, accurate mechanisms, which can ﬁnely feed, pick, orientate, move, and release diﬀerent types of objects at the right place. The assembling and packaging operations that achieve the microcomponents’ fusion into a hybrid microsystem is usually considered a bottleneck in the manufacturing process more than the manufacturing of components itself. This is particularly true for very small components that require high positioning tolerances leading to high manufacturing cost. High cost gripping solutions for various applications concerning the handling and the assembling of microcomponents have been developed but they do not oﬀer satisfying economical solutions yet. According to Breguet et al. [30], the main three challenges characterizing microassembly are the following: • • •

Precise alignment (submicron) of the components in several degrees of freedom and in a large workspace (a few cm3 ) Grasping and releasing of these delicate components Attaching them together

We present a taxonomy of microassembly in this chapter. To produce a miniaturized multifunctional system, we distinguish the following criteria: • •

Do we have to assemble a composed product or can we design it to avoid (or at least reduce) assembly tasks? Do we assemble a lot of loose components or can we combine assembly and manufacturing in situ? [167]

4

1 From Conventional Assembly to Microassembly Multifunctional product

Monolithic product

Composed product

Combined part manufacturing and assembly

Product external assembly functions

Assembly of loose components

Self-Assembly or stochastic assembly

Product internal assembly functions

Fig. 1.1. Taxonomy of microassembly

• •

Is the assembly equipment inside or outside the product? Can we use selfassembly (also called stochastic assembly)? [167] Finally, is the assembly required to be serial or can the throughput be increased by using parallel assembly? [22]

This classiﬁcation is shown in Fig. 1.1.

1.2 Design of Monolithic Products for Microassembly The ﬁrst and most basic approach for microassembly consists in downscaling the conventional approach. The use of miniaturized grippers (mainly downscaled tweezers or vacuum grippers, but an exhaustive description of the suitable gripping principles is presented in Chap. 3) allows to pick, move, orientate, and release microcomponents; however, the word gripper explicitly refers to a two ﬁnger tool used to grip an object, it must be understood here as any device allowing to pick a component, such as, for example, a vacuum gripper with only one ﬁnger. The most often associated strategy consists in serial pick and place of components. The main drawbacks of such an approach consist in physical limits (sticking problems at release) and in nonoptimal solutions (all eﬀorts for accurate positioning must be repeated for each component). Some authors [65] propose to improve this situation by combining design for both microassembly and microworld adapted assembly equipment. The design for microassembly is supposed to reduce the number of assembly tasks or at least to improve the suitability of design for automated assembly. To illustrate this, let us consider the design of a rotational joint with a one-way actuation and an elastic force to get the system back to the equilibrium. This example is illustrated in Fig. 1.2. In the conventional design, the rotational joint is made of a small ball bearing (SKF produces reduced ball bearing with

1.2 Design of Monolithic Products for Microassembly

5

Spring

Moving part

Moving part

Notch hinge

Ball bearing

Fig. 1.2. Conventional design vs micro-driven design: case of a rotational joint (the actuator is not shown)

outer diameter of about 2 mm. As a comparison, one of the smallest ball bearing with an outer diameter of 0.9 mm has been assembled by the researchers of the MEL1 (Japan) with their microfactory [6, 135]. Once both parts and the ball bearing are assembled, they still need to be put together with the actuator and the elastic element allowing the backward motion. Several elements have to be manufactured and assembled. Besides the hardness of the task, the diﬀerent tolerances lead to a low-eﬀective system with clearances. Moreover, if the moving part has to guarantee watertightness with an antagonist counterpart, the system will probably not meet the requirements. An alternative design could replace the ball bearing and the elastic element by an elastic ﬂexure hinge, combining both functionalities in one component. The maximal deﬂection of the notch hinge – a notch hinge is a ﬂexure hinge with a circular proﬁle – depends on the Young’s modulus and the elastic limit of the material, and on the width and thickness of the hinge [35]. With titanium and wire electro-discharge machined2 hinges with 5 mm thickness and 100 µm width, the angular range can reach 15◦ . This new design reduces the number of parts, highly simpliﬁes the assembly task, and enhances the functionality of the system: no clearance, no friction, no wear, and consequently no scraps, making this kind of design particularly suitable for biocompatible applications.

1

2

MEL = Mechanical Engineering Laboratory, see National Institute of Advanced Industrial Science and Technology http://www.aist.go.jp. EDM = Electro Discharge Machining is “a machining method using a free electrical discharge between an electrode and a workpiece to generate heat ﬂow with high energy density, so that contact force and chatter vibration can be avoided during machining” [87].

6

1 From Conventional Assembly to Microassembly Micro electro-discharge machining (drilling)

Insert of the pin

Twist for breakage Removal of neck material by of the neck micro-electro discharge machining

Workpiece Pin produced with wire electro discharge grinding

Ultrasonic vibration of the worktable

Fig. 1.3. Pin-in-hole combination produced by part manufacturing and assembly steps [115] (Copyrights CIRP.)

1.3 Combined Part Manufacturing and Assembly According to Tichem and Karpuschewski [166], “the goal of this method is to minimize the assembly content of composed products by creating products on basis of a combination of part manufacturing and assembly operations. This reduces the amount of part handling operations and delicate joining operations. Clearly, this method is not a pure assembly method. It recognises the fact that, in the microdomain, part manufacturing and assembly can in certain cases be integrated. The separation between part manufacturing and assembly as visible in the macrodomain vanishes.” An example of this approach can be found in [115], dealing with a pin-in-hole combination performed by part manufacturing and assembly steps. More recently, Jing-Dae Huang and Chia-Lung Kuo[87] have improved this method combining micro-EDM and laser welding to manufacture and achieve pin-plate assembly of 50 µm diameter pins with a large aspect ratio.

1.4 Product External Assembly Functions This approach is the most conventional one, often based on the use of accuracy positioning systems and microgrippers. This method can be improved by adding visualization systems and imaging processing. We will not focus on this method because it is not micro-oriented. Nevertheless, it has to be mentioned because it can lead to a micro-speciﬁc assembly method when coupled with a parallel assembly approach, which is also presented in this section.

1.5 Product Internal Assembly Functions The principle of internal assembly is to provide a microproduct with additional functionalities such as, for example, internal actuators. The assem-

1.6 Stochastic or Self-Assembly

7

Fiber holding groove Bimorph

Passive spring Silicon wafer surface

Fiber

V-beam (thermal actuation)

Fig. 1.4. Example of internal adjustment of optic ﬁbers (Reprinted with permission from [82]. Copyrights 2006 Institute of Physics.)

bly of this product with another component can then be performed in two steps: A ﬁrst coarse positioning of the component on the product is performed with a conventional, whether miniaturized or not, handling tool. The ultimate positioning with the required accuracy is performed inside the product, thanks to the internal actuators. This way to perform assembly provides ﬁnal microproducts with a higher complexity and more internal functionalities. The cost aspects of such a method must be analyzed carefully. The example of internal assembly given in [175] includes the following functions for a self-adjusting microsystem: a controlled actuation of the component, the sensing of the position of the component, and the freezing of the component in the ﬁnal position. The studied example consists in interconnecting optical ﬁbers with each other. The ﬁne positioning is performed by using a piezoelectric plate glued on a passive silicon layer, allowing a bending motion of the actuator when voltage is supplied to the piezoelectric electrodes. This bending motion is used for the ﬁne positioning of the ﬁbers. Two ways can be used to keep the alignment: The ﬁrst way is to use active control, the second one consists in permanently freezing the alignment, but nowadays, no technical solution has been proposed yet for this application. Recently, an adaptation of this principle has been proposed in [82], which is illustrated in Fig. 1.4.

1.6 Stochastic or Self-Assembly The underlying idea of stochastic assembly is to avoid any deterministic interaction and control of the part position during the assembly task. The components to be assembled are jumbled in close distance before applying a force ﬁeld that will perform the assembly. An example of stochastic assembly

8

1 From Conventional Assembly to Microassembly

Fig. 1.5. Example of stochastic assembly illustrated in [23]: capillary forces in the adhesive (black) cause self-assembly

is cited in [166] and consists in using the capillary force of an adhesive drop to align two parts (information on lateral forces modeling can be found in [105]). This principle taken from [23] is illustrated in Fig. 1.5. Other eﬀects are cited in [22] and have been applied by several authors to proceed stochastic assembly: • • • •

Fluidic agitation and mating part shapes Vibratory agitation and electrostatics force ﬁelds Vibratory agitation and mating part shapes Colloidal self-assembly [106]

1.7 Parallel Assembly Unlike the sequential process, parallel assembly is the assembly of more than two devices at the same time. As a comparison, batch fabrication is widely used in microelectronic where the same parallel concept is applied for silicon chip fabrications. Parallel assembly avoids the drawbacks of a sequential one for small devices: time consuming and low throughput. The drawback of parallel assembly is less ﬂexibility compared to sequential assembly. An example of parallel microassembly with electrostatic force ﬁelds is given in [22].

1.8 Conclusions These diﬀerent strategies for microassembly are logistic approaches for assembly (we have not discussed technology yet). It is now essential to focus on the ways to perform the assembly tasks: feeding, positioning, pick, and place. More speciﬁcally, we intend to focus on microgripping. Therefore, the following chapters are devoted to the forces acting at the envisaged scale (10 µm to 10 mm) and the related handling principles.

2 Classiﬁcation of Forces Acting in the Microworld

2.1 Introduction When downscaled, volumic forces (e.g., the gravity1 ) tend to decrease faster than other kinds of forces such as the capillary force or the viscous force. Although they still exist on a macroscopic scale, these forces are often negligible (and neglected) in macroscopic assembly. A reduced system is consequently brought face-to-face with the relative increase of these so-called surface forces. According to the literature on microassembly, these forces are mainly the electrostatic forces, the van der Waals forces, the liquid bridge (also called capillary or surface tension) forces, the forces due to the mechanical clamping (contact forces) and deformation (pull-oﬀ forces), and viscous drag. The term surface force is misleading since all these forces does not really depend on the square of the characteristic length. Nevertheless, this term conveys the idea that these forces decrease more slowly than the weight, which leads to some cut-oﬀ sizes below which these forces disturb the handling task because they generate the sticking of the microcomponent to the tip of the gripper (the weight is no longer suﬃcient to overcome them and ensure release). There are several ways to tackle this problem: These forces can be reduced, overcome, or exploited as a gripping principle. The choice will be diﬀerent according to the manipulation strategy (see Fig. 2.1): The parameters (materials, environment, geometries) will be chosen to maximize the force used as a gripping principle (for example by choosing hydrophilic materials in a manipulation based on the capillary force) and to minimize the disturbing forces (use of hydrophobic materials in a manipulation based on a mechanical gripper). This chapter presents some general classiﬁcations of the forces according to their range and introduces the most often cited forces in microassembly literature.

1

From one point of view, inertia forces also involve the mass of the component, but the possible high dynamics at small scales compensate this eﬀect, as illustrated by the dynamical release proposed in [77].

10

2 Classiﬁcation of Forces Acting in the Microworld Forces

Gripping principles

Fig. 2.1. Forces and gripping principles: The force underlying in a gripping principle must be maximized, all the others should be decreased

2.2 Classiﬁcation Schemes of the Forces According to Lee [118], we go over the ﬁrst simpliﬁed classiﬁcation of the diﬀerent forces in four main categories: • • • •

Gravity, with an inﬁnite range Electromagnetic force, with an inﬁnite range Weak force, with a range smaller than 10−18 m Strong force, with a range smaller than 10−15 m

These last two forces are outside the scope of this work due to their very short range (inside the nucleus). Electromagnetic forces represent the source of all intermolecular interactions and their inﬂuence can be combined to that of gravity in some phenomena such as the rise of a liquid in small capillaries. The interaction between atoms, molecules, and solids is characterized by the following: • •

Chemical forces and covalent bonding, with a range over the order of an interatomic separation (typically 0.1–0.2 nm) Coulomb force and ionic (or partially ionic) bond

Moreover, the interaction between microscopic bodies also depends on the Lifshitz–van der Waals (VDW) forces, which can be classiﬁed into four categories: •

Dispersion forces, also called London forces [120], are due to a Coulomb interaction. They represent one third of the Lifshitz–van der Waals forces, are long range (more than 10 nm), can be attractive or repulsive, and act between all atoms and molecules, even between neutral ones. These forces are nonadditive, which means that the interaction between two molecules is aﬀected by the presence of other bodies. The interaction energy of the dispersion forces decreases as a function of the separation distance to the sixth power ( r16 )

2.2 Classiﬁcation Schemes of the Forces

•

•

11

Orientation forces, also called Keesom forces, coming from the interaction between two permanent dipoles. Their energy also depends on the separation distance as r16 Induction forces, also called Debye forces, due to the interaction between a permanent dipole and an induced dipole, with an energy decreasing as 1 r6

•

Retardation forces, described by Casimir and Polder, due to the nonnegligible propagation time of the electromagnetic wave between the dipoles when their separation distance becomes higher than typically 10 nm. Because of this propagation time, the relative orientation of the dipoles are less favorable and the interaction energy decreases faster than for the other terms ( r17 )

A detailed description of these four terms can be found in [118] (Table 3, p10) or in [88]. At this stage of reading, it seems that the fast decrease of the van der Waals with the separation distance put them aside as far as microsystems are concerned. However, a more subtle investigation shows that this decrease complies with another power law in the case of two macroscopic bodies interacting with each other [89]. Still mentioned in [118], the Coulomb and Lifshitz–van der Waals forces are not suﬃcient to explain the adhesion between two solids: the molecular interactions (also called donor–acceptor interactions by physicists or acid–base interactions by chemists) also play a role in adhesion, but as their range is limited to the interatomic separation (typically smaller than 0.3 nm), we will not consider them in what follows even if a more detailed study concerning the close contact of two bodies should probably involve their eﬀects. Finally, we cannot conclude this section without mentioning the role of capillary forces [48], [80]. These forces play an important role in a lot of surrounding phenomena and applications: They allow children to build up sand castles and everyone to collect the crumbs more easily, provoke adhesion between microcomponents, cause reliability failure in MEMS2 applications [104, 125, 176, 179, 184], and are of the utmost importance in microassembly. As a ﬁrst conclusion, we propose the schematic summary presented in Table 2.1. Table 2.1. Forces summary according to the interaction distance Interaction distance Up to inﬁnite range From a few nm up to 1 mm >0.3 nm 0.3 nm < separation distance <100 nm <0.3 nm 0.1–0.2 nm

2

Micro Electro Mechanical System.

Predominant force Gravity Capillary forces Coulomb (electrostatic) forces Lifshitz–van der Waals Molecular interactions Chemical interactions

12

2 Classiﬁcation of Forces Acting in the Microworld

To make this ﬁrst classiﬁcation easier to use from a mechanical point of view, we have proposed [112] a diﬀerent classiﬁcation, making the distinction between forces at contact (forces including deformations – JKR, DMT, and related indicators,3 interaction energy of two bodies, and friction) and forces at distance (surface forces including van der Waals forces, electrostatic forces, and capillary forces). This classiﬁcation is valid for gaseous environment. On the other hand, the case of immersed environments is tackled in [64].

2.3 Conclusions The problematics of microforces has already been described by several authors. Maybe the most cited surveys in the microassembly literature are the works given in [28], [58], and [89], which summarize the most important forces acting when dealing with microparts: the electrostatic forces, the van der Waals forces, the capillary forces, the gravity forces, and the viscous forces. Although the way these forces are involved in microassembly is not completely understood yet, it is now well established by the scientiﬁc community that these forces are no longer negligible when manipulating and assembling parts within the size of 0.1 mm and smaller. These eﬀects are also experimented by a lot of industrials involved in the handling of components of watches or mobile phones [136]. Let us note that these disturbing side-eﬀects are not limited to assembly but are also encountered in manufacturing by microstereolithography: The breakdown of small mechanical structures due to the collapsing induced by the capillary forces (the forces arise from the presence of a rinsing liquid after the polymerization phase of the process) is reported in [184]. Some authors suggest to assemble component in a liquid environment: Gauthier [64] has recently explained how the above mentioned forces were decreased in water4 . This current research ﬁeld on working in liquid media is beyond the scope of this book. In Chap. 3, it will be focused on the way these forces have been used through the literature as handling principles.

3

4

These models are extensions of the Hertz model, which takes adhesion into consideration. The Johnson–Kendall–Roberts model is more adapted for high adhesive or low stiﬀness contacts while the Derjagin–Muller–Toporov model is more adapted to low adhesive or stiﬀ contacts. Capillary forces are totally suppressed since there is no longer a liquid/gas interface, van der Waals forces are decreased because the so-called Hamaker constant (which is proportional to the force) is usually smaller in liquid environments, and electrostatic forces are decreased since the dielectric constant is 80 times larger for water than for vacuum.

3 Handling Principles for Microassembly

3.1 Introduction Theoretical classiﬁcations of forces acting at the considered scale have been presented in the previous chapter. Now, we summarize the handling principles which have been proposed in literature in order to transform these forces in technological solutions. As a lot of diﬀerent gripping principles exist, it has been decided to put forward (1) a ﬁrst overview (based on a compilation of existing [2, 141] and own [113, 177] states of the art), (2) an own classiﬁcation scheme, and (3) the comparison scheme proposed by Tichem et al. [169], used at the end of the chapter.

3.2 Presentation of Gripping Principles A ﬁrst overview of the scientiﬁc literature on the gripping principles (not particularly the “micro” ones) leads to Fig. 3.1. Let us now enumerate the most common principles: 1. The friction based gripping using miniaturized tweezers [2, 102] (this is probably the most widespread gripper in industry, together with the vacuum gripper presented in what follows); to illustrate the mechanical gripper based on the tweezers principle, we present a microgripper with two ﬁngers actuated by piezoelectric bimorphs (Fig. 3.2). The whole gripper is packaged like an electronic chip. This is an example of “plug and use” concept developed by the LAB (Laboratoire d’Automatique de Besan¸con, [2]). The ﬁngers have been manufactured by LIGA process1 . Figure 3.2b 1

“The LIGA process was developed at the IMT (Institute of Microstructure Technology), in the early eighties under the leadership of Dr. W. Ehrfeld. LIGA is an acronym standing for the main steps of the process, i.e., deep X-ray lithography, electroforming, and plastic molding (LIGA means Lithographie– Galvanoformung–Abformung). These three steps make it possible to massproduce microcomponents at a low-cost,” Source: http://www.fzk.de.

14

3 Handling Principles for Microassembly

+++ ----(a)

(b)

(f)

(c)

(g)

(k)

(h)

(l)

(d)

(i)

(e)

(j)

(m)

(n)

Fig. 3.1. Several gripping principles: (a) Tweezer or friction based gripper; (b) Form closure gripper; (c) Vacuum gripper; (d) Magnetic gripper; (e) Electrostatic gripper, (f ) Push–pull gripper; (g) Capillary or surface tension gripper; (h) Ice or cryogenic gripper; (i) Bernoulli gripper; (j) Air cushion handling system; (k) Standing waves gripper; (l) Squeeze ﬁlm gripper; (m) Optical gripper, and (n) van der Waals gripper. (Drawings a,b,c,e, and i are taken from [168], Copyrights CIRP.)

(a)

(b)

Fig. 3.2. The plug and produce concept developed by the LAB: the microrobot on chip (MOC). (a) The two-ﬁnger gripper on chip; (b) gripping of Φ 0.2 mm watch components (Courtesy of Jo¨el Agnus, Laboratoire d’automatique de Besan¸con.)

3.2 Presentation of Gripping Principles

15

Table 3.1. Properties of the two ﬁnger grippers: the gripping amplitude is the amplitude of the motion in the plane of the ﬁngers, the insertion amplitude is that of the motion perpendicular to this plane Gripper Actuation Fingers Gripping

Tip

Parameter Piezoelectric bimorphs Length Material Gripping amplitude Insertion amplitude Gripping force Insertion force Length Height

Value

Unit

25 Ni 320 200 80 30 1 0.3

mm 10−6 m 10−6 m 10−3 N 10−3 N 10−3 m 10−3 m

Base cantilever

Open

P+dP 200 – 400 µm

Close

Micro container

(a)

Rubber Silicon

P

(b)

Fig. 3.3. Example of form-closure gripping of a microbe [134]. (a) Schematic view of cage operation: The cage opens and approaches the microbe, then closes and captures the microbe; (b) Pneumatic actuation: Flexure of the rubber membrane by pressure causes the ﬁngers to tilt, creating an opening for object entry ([134], ©1999 IEEE)

shows the gripper handling small watch components (the typical diameter of the gear shaft is about 0.2 mm). The parameters of the gripper shown in Fig. 3.2a are summarized in Table 3.1. As a new development of this kind of gripper, an automatic tools changer has been proposed in [39] order to make this gripper suitable for microassembly in SEM2 environment. 2. The form closure gripping This principle is presented in [134], who uses it to handle sensitive elements like microbes. The working principle is illustrated in Fig. 3.3. 3. The vacuum gripper [45, 51, 163, 143, 157, 189, 190]. The pressure diﬀerence between the ambient atmosphere and the “vacuum” generated inside 2

SEM: scanning electron microscope (the sample has to be put in a vacuum chamber).

16

3 Handling Principles for Microassembly

Hollow-bored sonotrode Vacuum

Fv Fsf

Fsf

Levitated component Fig. 3.4. Example of vacuum gripper: A hollow-bored sonotrode pulls the component upwards (thanks to the vacuum suction) and repels it downwards (thanks to the squeeze ﬁlm eﬀect) [189] (Courtesy of Fraunhofer IPT.)

the gripper can be used to pick up microcomponents. Such tools are widespread in industry and an example of vacuum gripping tool can be found in [190]. It consists in a glass pipette and a computer-controlled vacuum supply. Because of the adhesion forces, pick operation and place operation have antagonist demands: The ﬁrst one requires a large tip diameter while the latter needs a small one. Thus, for each size of component, there is an optimal diameter for the pipette tip. For instance, when handling a 80–150 µm sized object, the best results were obtained with a tip size ranging from 25 to 50 µm, which is about 25–50% of the object size. This glass pipette is able to perform pick and place operations of 50–300 µm sized metallic and nonmetallic particles with a success rate of about 75%. The pump works with a 6 bar pressure supply and a voltage of 6 V. The maximum output vacuum is −0.86 bar and the maximum output pressure is 6 bar. The vacuum gripper presented in Fig. 3.4 is a combination between an attractive suction force and a repulsive squeeze ﬁlm eﬀect used to prevent the component from touching the gripper (see later, “Ultrasonic levitation,” p.23). In absence of this repulsive squeeze ﬁlm, the high contact forces during the picking phase can lead to the formation of cracks at the surface, as illustrated in Fig. 3.5. 4. The magnetic gripping, and particularly the magnetic levitation in which the force comes from a magnetic ﬁeld generated by magnets (Fig. 3.1d). Three diﬀerent types can be used: permanent magnets, electromagnets, or superconducting magnets. The use of electromagnetic levitation is limited to materials with high electrical conductivity and to low-temperature applications [96, 128, 140]. Recently, magnetic levitation has also been

3.2 Presentation of Gripping Principles

17

Vacuum nozzle

Crushed circular shape

Propagated crack

Fig. 3.5. The mechanical damage at contact (Courtesy of Assembl´eon.) Dm

Lm

V D

Lp Dp

Fig. 3.6. The electrostatic handling (Reprinted with permission from [151]. Copyright 2003, American Institute of Physics.)

used by [27] to propose new force sensor for microassembly, with a stiﬀness of about 0.02 N m−1 . 5. The electrostatic gripper , including the electrostatic levitation [52, 53, 56, 58, 83, 171]. A ﬁrst example of a micromanipulation task driven by the electrostatic eﬀect is presented in [151] and depicted in Fig. 3.6. The authors proposed to use the electrostatic force in order to handle (=catch, move, release) a spherical particle near a substrate plate. All the objects (gripper, sphere, and substrate) are conductive. The principle in this case is to use the adhesion force to perform the gripping task and to impose a detachment voltage for release. The adhesion is modeled according to the JKR model [93, 94] and the electrostatic force

18

3 Handling Principles for Microassembly

Electrodes

Top view

Side view

D

Fig. 3.7. Electrodes of the electrostatic gripper (Courtesy of J. Hesselbach, TU Braunschweig, [83].)

is computed by the boundary elements method. The eﬀects of roughness, the use of nonconductive materials, the electric discharge, and the tunneling current have not been taken into consideration. The results are given as a function of the geometry ratio, and seem to be independent from the actual size. A second example, a micromanipulation driven by the electrostatic force is presented in [83] and shown in Fig. 3.7 (data are given in Table 3.2). The authors described the handling of insulating objects: the gripping, the centering, the moving, and the releasing tasks are performed by controlling the electrostatic force. It turns out that the humidity rate greatly inﬂuences the reliability of this kind of electrostatic gripper. Above 65% RH3 , objects with a size >400 µm cannot be gripped, even at maximum voltage (1, 200 V). As the adhesion inﬂuences the releasing task, the reliability of this kind of gripper becomes problematic. As far as the electric levitation is concerned, diﬀerent kinds of particles can be manipulated such as, for example, conductive, semiconductors, and dielectric materials. A distinction can be made between the following: (a) Electrostatic. Static electrical ﬁelds can be used to levitate uncharged small particles by induced polarization of the sample. This technique is suitable only for polar liquids and low temperature ﬁelds. At high temperature the static charges are not maintained and gradually degenerating over time. On the other hand relatively large particles can be levitated [56, 92]. (b) Electrodynamic. The particles are charged and held stationary using a combination of static (DC) and oscillating (AC) electric ﬁelds. This technique is able to suspend and levitate small charged particle. The disadvantages lie in the poor particle stability and in the limited particle size (up to 200 µm) [11]. 3

RH = Relative humidity.

3.2 Presentation of Gripping Principles

19

Table 3.2. Properties of the electrostatic gripping [83] Part Substrate

Parameter Proﬁle Roughness Material

Handled components Spheres Material Diameter Cubic valve ﬂap Material Characteristic size Working conditions Voltage Humidity rate Electrode Inner diameter Outer diameter Gripping force (an estimation)

Value Unit Flat “smooth” Pyrex wafer Glass 100–800 Insulator 80 300–1200 <65 120 560 50

10−6 m 10−6 m V % 10−6 m 10−6 m µN

α

L h R l

D

(a)

(b)

(c)

Fig. 3.8. The push–pull manipulation. (a) Geometrical data; (b) example of tip: CS21 (Courtesy Mikromasch.); (c) example of tip: NSC12 (Courtesy Mikromasch.)

Finally, let us cite two additional devices that are currently being studied to achieve feeding and centering of millimetric components: Fantoni and Santochi [57] have developed a electrostatic feeder and Porta and Santochi [144] are currently developing a centering device based on electrostatic ﬁeld gradient (components are attracted toward high gradient area). This kind of devices already give satisfactory results for conductive parts but still suﬀer from a lack of precision as far as nonconductive materials are concerned. 6. The “push-pull” or adhesion based handling (Fig. 3.8) has already been described for example in [152]. It consists in pushing small spheres with the tip of an AFM4 (Atomic force microscope) located on a “plane” surface (Fig. 3.8a). Adhesive forces can prevent the separation of the tip from the sphere, leading to “adhesive pulling.” However, in some cases the release task can be performed, but a detailed study of the phenomena acting at 4

For an introduction to the atomic forces microscopy, see [31]

20

3 Handling Principles for Microassembly

Table 3.3. Properties of the push–pull handling Part Substrate

Handled sphere Cantilever

Tip Working space

Parameter Proﬁle Roughness Material Diameter Material Length (L) Height (h) Tilt angle (α) Height (l) Radius (R) PI Range: X,Y ,Z AFM Range: X and Y AFM Range: Z

Value Flat Unknown Glass 50 Polystyrene 350 2 30 10 10 15 × 15 × 15 45 4

Unit

10−6 m 10−6 m 10−6 m

◦

10−6 m 10−9 m mm3 10−6 m 10−6 m

the interface sphere-tip helps to understand and control this manipulation: Details of this manipulation are given in Table 3.3. The cantilevers are commercialized on http://www.spmtips.com/, and two cantilevers have been used: the CSC21 and the NSC12 (Fig. 3.8b and c). Another team, the Laboratoire de Robotique de Paris (LRP), has studied the adhesion based micromanipulation and has proposed related release strategies (by rolling, scrapping, and imposing high dynamics). A pointer to their work can be found in [76]. 7. The capillary or surface tension based gripper , in which the surface tension forces can be used to get parts stuck to the gripper [12, 13, 26, 72, 108, 109, 112, 138, 153]. [13] used a low viscosity liquid such as ethanol that evaporates without leaving particles on the part’s surface. Several parameters intervene in the accuracy with which the part can be handled, such as the gripper’s shape, the part’s shape, and the diﬀerence of size between the gripper and the component. The so-called centering eﬀect occurs and causes an orientation of the component at the gripper’s shape. The lifting force depends on the type of adhesive, the gripper’s surface, the gripping distance, the adhesive volume, and the materials. The second example [72] of capillary gripper was used to manipulate small Si plates of 2 × 2 mm2 . Authors of [26] suggest to vary the gripper curvature in order to vary the capillary force, according to the well-known Israelachvili approximation (18.2). Therefore, as illustrated in Fig. 3.9a, the extremity of the gripper is made of a deformable membrane whose curvature increases under the action of a liquid (i.e., this liquid is inside the gripper, and only used to drive the shape of the membrane. It can be totally diﬀerent from the liquid constituting the meniscus between the gripper and the component). Nowadays, the main limitation of the method is its diﬃculty to be miniaturized. Recently, Pagano et al. [138] has proposed a new design (Fig. 3.9b)

3.2 Presentation of Gripping Principles

21

Actuation volume

First electrode

R Zero curvature

Curvature = 2/R

(a)

First electrode Actuation volume Squeezed EAP

Non squeezed EAP

Second electrode

Second electrode

(b)

Fig. 3.9. Some examples of surface tension based gripper designs. (a) Variable curvature driven by a liquid; (b) variable curvature driven by an actuated EAP [138] (Copyrights CIRP.)

where the deformation of the membrane is obtained by using an electronic EAP5 between two electrodes which can squeeze it to provoke a change of the curvature of the gripper tip. This principle is still to be implemented. Saito et al. [153] has proposed to use a gripper with a hemispherical concavity in order to increase the contact conformity between the gripper and a spherical component (Fig. 3.10a). This allows to pick it up from a ﬂat plane. To achieve the release, the authors propose to increase the volume of the liquid, which leads to a force decrease. This principle has still to be implemented. 8. The ice or cryogenic gripper , also called ice gripper [131, 101, 114, 158]. A gripper that was developed in the context of an Eureka project, including the participation of the Swiss Centre for Electronics and Microtechnology (CSEM), AP Technologies, and Sysmelec, is described in [131]. This method is based on the adhesive properties of ice to pick up the microparts. The gripper developed by the CSEM6 ﬁrst sprays a drop of water onto the object and then it gets close to the object until it touches it. As soon as there is a contact between the gripper and the object, water freezes and the component is stuck to the gripper so that it can be manipulated at will. It should be noted that the gripping strength of ice is 20–100 times stronger than that obtained with vacuum grippers. To release the object, the tip of the gripper is simply warmed up to the phase change temperature of the liquid interface. A prototype was able to handle components sized between 0.1 and 5 mm, with an accuracy of 1 µm and a rate of 1,000 cycles per hour. The advantages of the cryogenic gripper are as follows: High adhesion forces, the surfaces of the object are not damaged during the handling process, the handling process is almost independent from materials properties, and short pickup and release times [158]. An example of millimetric component hold by a cryogenic gripper is given in Fig. 3.10b. 5 6

EAP = Electro Active Polymer. Source: http://www.devicelink.com/emdm/archive/98/09/tech.html.

22

3 Handling Principles for Microassembly

Gripper Component Plane

(a)

(b)

Fig. 3.10. (a) Examples of surface tension gripper based on conformity to increase the capillary force (Reprinted with permission from [153]. Copyright 2005, American Institute of Physics.); (b) Example of cryogenic gripper (Courtesy of Defeng Lang, TU Delft.) Table 3.4. Examples of phase changes proposed by [114] as gripping principles Intermediate Water Magnetic-rheological ﬂuid Thermoplastic polymer Thermoset polymer

Process Heating–Cooling Electromagnetic ﬁeld Heating–Cooling UV based principle

A new adaptation of this principle has recently been proposed in [121], using a submerged gripper. In this case, the liquid environment ensures a low adhesion between the gripper and the component and a rigid link between them can be created or suppressed, thanks to the control of a Peltier element. Note that the cryogenic gripping principle has been generalized under the terminology of phase changing gripping [114], whose several examples are given in Table 3.4. 9. The aerodynamic levitation: according to the direction of the ﬂow, two diﬀerent approaches can be used: (a) Bernoulli levitation. The sample is held below the manipulator through which air ﬂows downward (Fig. 3.1i). Because of the high pressure supply, air radially ﬂows between the gripper and the component. The velocity increase induces a dynamic pressure decrease (Bernoulli eﬀect), leading to an upwards attracting force on the component [54, 72, 180]. (b) Air cushion levitation. The sample is held above the manipulator (Fig. 3.1j). Pressurized air ﬂows upwards through several holes that are drilled all over the gripper and leads to a repulsive levitation force that counterbalances the weight of the component [66].

3.2 Presentation of Gripping Principles

23

Reﬂector z

z

z

+ g Heavy sphere

n λ2

Stable Unstable

+

-

Pressure

Velocity

Piston sound source

Force

Fig. 3.11. Levitation of dense spheres in an acoustic standing wave [147] (Copyrights CIRP.)

Recently, Nyhuis and Fiege [132] have proposed a feeding concept using aerodynamics orientation methods. According to the authors, these latter fall into two categories: passive orientation methods, where incorrectly oriented parts are rejected by means of aerodynamic baﬄes, and active orientation methods, where the workpiece is actively moved into the desired position. 10. The ultrasonic levitation, divided into the following: (a) Standing waves levitation. Small components can be levitated in the pressure nodes of an acoustical standing wave between a vibrating plate and a reﬂector (Fig. 3.11). Moreover, because of the pressure distribution, a radial ﬂow, whose velocity decreases toward the periphery, induces radial centering by Bernoulli eﬀect reaching up to 30% of the levitation force [14, 38, 40, 43, 68, 74, 86, 99, 148, 186, 188]. The company Dantec Dynamics7 can be asked for information concerning an ultrasonic levitator [44]. (b) Squeeze ﬁlm levitation or near ﬁeld levitation. The reﬂector of the standing wave levitation is replaced by the levitated object (Fig. 3.1g). Consequently, any weight can be levitated if the separation distance between the object and the vibrating plate is small enough (see the typical proﬁle of force as a function of separation distance in Fig. 3.12). This technique is often referred to as near ﬁeld levitation according to several authors [86, 78, 172, 183]. The combined use of vacuum attraction and squeeze ﬁlm repulsion has already been illustrated in Fig. 3.4. 7

http://www.dantecdynamics.com.

3 Handling Principles for Microassembly Levitation force

24

Near-ﬁeld levitation

Levitation distance λ/2

λ/2

λ/2

Fig. 3.12. Levitation force as a function of levitation distance [147] (Copyrights CIRP.) Beam axis z

Light intensity

i FD

a

o FD

FRi

Beam radius r

B

f

FRo

b

(a) Fig. 3.13. Optical forces acting on a sphere (Reprinted ﬁgure with permission from [7]. Copyright (1970) by the American Physical Society.)

11. The optical or laser gripper [5, 7, 8, 10, 9, 122, 146, 164]. Because of both beam reﬂection and refraction, the component undergoes an axial force that always pushes it forward in the direction of the beam and a radial gradient force that traps it in the center of the beam. This eﬀect can be understood as follows for a sphere with higher refractive index than that of its surrounding medium (water, oil): As depicted in Fig. 3.13 the rays a, B, and b undergo both reﬂection and refraction (also called deﬂection) at input and output interfaces. Because light carries momentum, the changes in ray directions cause (and are caused, i.e., the action/reaction principle

3.3 Classiﬁcation of Gripping Principles

25

is applied) the forces FRi and FRo due to reﬂection and the forces FDi and FDo due to deﬂection. These forces accelerate the sphere in the +z direction of the beam. For the stronger ray a, FRi and FRo are balanced to ﬁrst order while FDi and FDo radially add in the −r direction, leading to a net inward radial force (the beam axis z represented in Fig. 3.13 is the center of the beam). For similar reasons, the weaker ray b leads to a radial outward force of smaller magnitude. Therefore, the sphere is trapped in a stable way in the center of the beam. Nevertheless, it is preferred [146] to use this principle in a liquid environment because the condition on refractive index is respected and because this medium damps the oscillations of the component. This technique is restricted to very small transparent dielectric samples. Note that this principle leads to a gripping force that hardly reaches 1 nN [141], which represents the weight of a 33 µm edge cube with a density of 2700 kg m−3 (i.e., aluminum component). 12. The gripping based on van der Waals forces has been proposed in [4, 59] but no prototype was built. Actually, it seems that the adhesion based gripper probably uses van der Waals forces, combined with surface tension or electrostatic eﬀects.

3.3 Classiﬁcation of Gripping Principles We classify the handling principles according to the way they tackle the surface force disturbing problematics arising from the downscaling. We distinguish four strategies as indicated in Fig. 3.14. The ﬁrst two strategies consist in downscaling the existing equipment and in proposing some solutions to reduce or overcome the sticking aspects. The last two solutions are more microassembly oriented in the sense that they are based on strategies taking advantage of downscaling (typically these principles will probably not be adapted to the manipulation of conventional macroproducts) or avoiding any contact.

Microassembly

Feeding

Handling & Orientation

With contact

1 - Reduction

2 - Overcoming

Joining

Without contact

3 - Exploitation

4 - No surface forces

Fig. 3.14. Four strategies as far as surface forces are concerned

26

3 Handling Principles for Microassembly

1. Downscaling Approach I: How to Reduce the Surface Forces Eﬀects? The ﬁrst way to deal with these surface forces is a downscaling approach that consists in performing assembly on a conventional way with downscaled grippers. To avoid or to reduce the sticking eﬀects, the surface forces eﬀects are reduced by an adapted choice of the manipulation parameters. To decrease van der Waals forces, it is suggested to • Decrease surface pressure (=gripping force), to keep the contact area as small as possible • Prefer hard materials, to keep the contact area as small as possible [58] • Increase roughness [3, 58, 178, 192] • Control roughness proﬁle [107] To decrease the surface tension eﬀects, it is proposed to • Reduce the number of contact points • Use a microheater to evaporate the water bridge [3] • Use anti-adhesive, hydrophobic coating [3] • Handle in a liquid medium [170, 181] • Work with a low humidity rate [58, 80, 192] • Work under vacuum To reduce the electrostatic force, it is proposed to • Ground conductive materials [59] • Use materials with a small contact potential diﬀerence [58] • Use conductive materials that does not easily form insulating oxides [58] • Ionize the surrounding environment 2. Downscaling Approach II: How to Overcome the Surface Force Eﬀects? This second downscaling approach summarizes several techniques proposed by authors confronted with sticking eﬀects during the release task. These propositions are far from being solutions, because they come from a narrow point of view and cannot result from an integrated approach. For example, it has been suggested to work in dry air environment to avoid the formation of liquid bridges, but the breakdown voltage is then increased leading to larger electrostatic forces. However, these suggestions represent practical solutions for those who are confronted to sticking problems. General solutions (independently of the used gripping principle) suggest the following: • Use a large probe to pick up the micropart by adhesion and a smaller probe to release it (a smaller probe reduces the contact area and consequently, the gravity becomes dominant again and the object will stay in place when removing the small probe) [12, 190] • Change the pressure by controlling the temperature [3]

3.3 Classiﬁcation of Gripping Principles

27

• Use or, even better, control the adhesion between the handled component and the substrate • Improve the previous solution by joining the component to the substrate at the right place (for example by gluing) [21], [12] • Induce a relative motion between the component and the gripper by stripping oﬀ the component on a sharp edge [190] • Use vibrations to perform the release task [22] • Inject gas: A small puﬀ of gas pushes the object while removing the gripper [12, 190] • A similar approach to the vibration method is the dynamic approach that consists in communicating to the gripper an acceleration bigger than that the sticking force can communicate to the component [75, 149] Gripping principle related solutions consist in the following: • Using a positive pressure (vacuum gripper) [190] • Modifying the relative orientation of the component and the gripper [97] (when dealing with van der Waals forces, [59, 107]) • Destructing the gripping mechanism (for instance, with a gripper using the surface tension force to pick an object, the object can be released by heating the gripper and evaporating the adhesive liquid. Another example is the ice microgripper described in [131]) • Several propositions that still have to be validated, related to the capillary force: see later 3. Microworld Driven Approach I: How to Take Advantage of the Downscaling? Although adhesion can be reduced or overcome in some cases, it is sometimes interesting to exploit the surface forces eﬀects in order to perform the handling task. This approach, made possible by the interesting ratio surface forces/component weight, is followed by several authors who propose a lot of gripping principles related to microhandling: • Surface tension eﬀects • van der Waals force • Cryogenic gripping • Laser gripping • Bernoulli eﬀect • Handling in a ﬂuid medium [187] 4. Microworld Driven Approach II: Handling Without Contact Finally, these unavoidable diﬃculties lead to a fourth approach: If the gap between the component and the gripper can always remain larger than the cut-oﬀ lengths of the physical principles leading to adhesion, the handling task can be performed without paying attention to the surface forces (typically, this cut-oﬀ length is 10 nm for van der Waals forces [89]). The advantages of this noncontact handling approach can

28

3 Handling Principles for Microassembly

be summarized as follows. Some advantages cited are not limited to the handling of microcomponents. • Surface forces can be completely neglected • The friction eﬀect is drastically reduced, which enables high resolution and accuracy motion devices [96] • Handling of tricky (fragile, freshly painted, sensitive, or micron-sized structured surfaces) components is made possible because high local contact pressure by direct mechanical contact is avoided [189]. Handling of nonrigid products is also possible due to the ﬁeld of force [54] • Contamination from and of the end-eﬀector (in food handling [54] or in presence of lubricant [140]) can be totally avoided • In materials science, measurements of some physical properties are allowed avoiding undesired contamination from the container and eliminating wall-driven heterogeneous nucleation [63] The literature review highlights ﬁve distinct levitation techniques: (a) Magnetic levitation (b) Electric levitation (c) Optical levitation (d) Aerodynamic levitation (e) Ultrasonic acoustical levitation Once the object is levitated, another principle can be used for the horizontal displacement of the component ([66] therefore uses the electric ﬁeld) or the whole levitation system can be moved. The control of the levitation height should also give an additional degree of freedom when handling the component. Additional information on this topic can be found in [113, 177].

3.4 Comparison between Gripping Principles To compare all these principles, a classiﬁcation scheme is required. A modiﬁed version of an already proposed scheme [169] takes the following aspects into account: • • • • • • • • •

Material type of the handled component (for example, dielectric properties, porosity) Surface properties Speciﬁc grip force Force control aspects Remarks on possible limitations Accuracy Sensitivity to adhesion Environment Downscaling law

3.5 Conclusions

• •

29

Applications ﬁeld Handled components

Let us emphasize the fact that the required information is not available in all cases, leading to empty cells in Tables 3.5, 3.6 and 3.7. Note that all the aspects related to the component are listed in the norm DIN 325638 (Production equipment for microsystems–System for classiﬁcation of components for microsystems). Other classiﬁcation schemes could be based on diﬀerent criteria. For example, the actuation type can help to distinguish diﬀerent miniaturized tweezers. Moreover, a lot of grippers based on several technologies have been designed: piezoelectric actuated [29] microgrippers, thermally actuated grippers [119], SMA9 actuated grippers [17], piezoelectric bimorph gripper [2]. Other grippers already integrate a force feedback: gripper with an integrated piezo-resistive force sensor in the range of 1 µN [191], gripper with an attached strain gage [97], AFM force measurement in the range of 1 nN.

3.5 Conclusions In conclusion of this chapter on forces and gripping principles, we note that not all presented principles can be used to handle components with a size in the range between 10 µm and 1 mm. For example, the laser gripping principle must be put aside because of the low force it develops (0.7 nN). The cryogenic and the Bernoulli principles seem too “exotic” and their study does not help to understand the underlying phenomena mentioned in the microassembly literature. As far as the other principles are concerned, we preferred to focus on microworld driven gripping principles (van der Waals, electrostatic, surface tension forces). Note that the vacuum gripper is an interesting alternative to the downscaled mechanical gripper (it allows the handling of plane components, which is impossible with a two-ﬁnger gripper if the lateral faces of the components are not high enough). Nevertheless, the surface tension based gripping seems to be more promising (for example, in terms of downscaling laws) as presented in Chap. 6. Moreover, the study of the capillary force is a good way to enter the microworld because it is involved at many scales from the nanoscale (capillary condensation) up to the submillimetric scale. This will be discussed again in the chapter devoted to the capillary forces. 8

9

This norm takes the following parameters into account: geometrical shape, length, width, height and mass of the microcomponent, the shape (for example, a hole) and proﬁle (for example, convex surface) of the surface available for handling, the number of available faces available for handling [...], material characteristics and sensitivity to environmental conditions (for example, UV radiation), mechanical properties, roughness characteristics (Ra ), physical properties, clean room related aspects. Shape Memory Alloy.

Material limitation None

Surface properties Force 0.1 µN [98], 1 mN [69], 2–5 N [141]

None

None Condition on the refractive index [7]

11 Standing waves

12 Squ. ﬁlm 13 Optical

Not limited 0.1–10 pN

150 µN [63]

5 Electrostatic Better control with Rough surface Not limited conductive materials is a drawback [167] 6 Push-Pull 7 Capillary Hydrophilic, oleophilic [182] 8 Cryogenic None [167] 1 Nmm−2 [167] 9 Bernoulli Not too compliant, 0.1–10 N not too porous [54] 10 Air cushion None

2 Form closure None 3 Vacuum Limits on Contact may porosity [167] cause damage [167] 4 Magnetic Ferromagnetic [167] Not limited

Principle 1 Friction

Table 3.5. Comparison between the gripping principles Remarks At least 2 accessible surfaces [167]

Euler instabilities (plate does not remain parallel to the nozzle surface) Horizontal instabilities due to the absence of centering eﬀect [66] Axial stabilization [16] Orientation systems [15] Centering eﬀect [43] Centering eﬀect [79]

Only planar objects Damage due to radiation pressure[10]

Reduced ergonomy due to the reﬂector

External pressurized air supply is needed

External pressurized air supply is needed

Open (closed) loop for transfer (positioning)[140] Disturbing forces RH should be due to triboelectriﬁcation <40–60%

Control Important for fragile parts [167]

30 3 Handling Principles for Microassembly

Accuracy in the gripper part relation Not simultaneous contact of ﬁngers may introduce errors [167]

8 9 10 11 12 13

Cryogenic Bernoulli Air cushion Stand. waves Squeeze ﬁlm Optical

6 Push-Pull 7 Capillary

5 Electrostatic

4 Magnetic

Environment

No No No No

contact contact contact contact

Not in vacuum Not in vacuum Not in vacuum Not in vacuum Should be in a liquid

Not in a liquid

Depends on contact area Not in vacuum and surface properties Blow away is a release method [167] With or without contact

Sensitivity to adhesion

Triboelectriﬁcation may be a problem Adhesion is the working principle Automatic centering [141] Adhesion is the working principle 20 µm [182]

2 Form closed 3 Vacuum Relatively inaccurate [167]

1 Friction

Principle

Table 3.6. Comparison between the gripping principles

L [141]

L2 [141] (limit = max magnetic induction) L2 [141] (limit = breakdown voltage)

L2 [141] (limit = suction force)

L2 [141] (limit = strength)

Downscaling law

3.5 Conclusions 31

Applications

13 Optical

12

10 11

8 9

Crystals manipulation [10]

Handling of components with only one accessible surface Ball placement for a watch ball bearing Cryogenic Clamp before machining Bernoulli Food handling [54] Semiconductor devices handling [18, 123] Air cushion Microparts transfer system [66] Stand. waves Crystals manipulations [63] Small part handling in a commercialised device [160] Squeeze ﬁlm Wafers transfer [148]

7 Capillary

6 Push–Pull

2 Form closed 3 Vacuum Typical in SMD components handling 4 Magnetic Containerless crystal growth [128] Wafer transfer [140] 5 Electrostatic Aligning, positioning and transferring components [56]

Principle 1 Friction

Planar objects (Φ 200 mm wafers) [148] Bakelite plate (90 mm × 65 mm2 , 8.6 g) [78] Spheres (5 µm), bacteria, cells [5] Silica particles (25 nm–10 µm) in water [8] Crystals seeds (1–25 µm) [10]

Aluminum blocks (65–130 g), jelly blocks (35–175 g) [54] Flat disk (R up to 150 mm and m up to 2 kg [180] Millimeter sized polymer plate (PMMA) [66] Φ 3 mm spherical crystals [63] Microgears (wheel Φ 3 mm, shaft Φ 0.8 mm) [85]

0.3 mm and 0.5 mm diameter balls [112]

Water, ice, sugar, quartz [128] 8 in. wafer [140], 4.2 mm × 4.2 mm2 components [182] Metallic cylinders (Φ 0.25 − 1 mm and 1–4 mm length) [56] Glass spheres (Φ 100–800 µm) [83] Not suited for sensitive IC components [141] Typically micrometric sized components like small spheres (polystyrene, glass) 2 × 2 mm2 Si components [70]

Handled object 2.7 µm polystyrene spheres, red-blood cells, protozoa [98], Φ 1 mm glass balls [165] Microbe [134]

Table 3.7. Comparison between the gripping principles (continued): examples of applications ﬁelds and handled components (SMD states for Surface Mounted Device)

32 3 Handling Principles for Microassembly

3.5 Conclusions

33

Electrostatic forces are probably a good candidate to develop microhandling tools, but we decided to put them aside because of the electromagnetic perturbations they induced on microcomponents, which often include electronics. Additionally, since they are long range, electrostatic force are suspected to disturb the manipulation task in case of electrostatic actuation of a two-ﬁngered gripper. Therefore, despite its theoretically less favorable response time, thermal actuation is currently envisaged as an alternative to actuations using a high electric ﬁeld. In what follows, a preliminary theoretical study has been led to assess the role of capillary forces. The ﬁrst reason for a theoretical modeling is that even if these eﬀects can be measured with adapted force sensors such as AFM, they cannot be separated from other contributions (all forces contribute to the global measured force amount). A model is therefore necessary to discriminate each contribution. The second justiﬁcation is that we intended to investigate the inﬂuence of a large number of parameters before setting up a experimental validation. Consequently, these eﬀects have ﬁrst been assessed by simulation. The third reason is that a simulation tool could help to better design microgrippers.

4 Conclusions

As it has been shown throughout this ﬁrst part, many current researches focus on grippers miniaturization. At scales larger than typically 1 mm, the reference gripping technique for components (typically SMD1 components) is the vacuum gripping. This technique is also applied for submillimetric applications; however, it seems to reach its limits2 . Therefore, it turns out from this literature review that at the considered scale (0.1–1 mm), eﬀects due to both gravity (depending on the cube of the characteristic size of the component) and vacuum suction (depending on the section area of the gripper, i.e., on the second power of the characteristic size) become overcome by capillary eﬀects. Indeed, although the capillary forces are often referred to as surface forces, they actually vary linearly as a function of the characteristic size, as it will be shown in Sect. 6.6. Consequently, scaling laws are more favorable to a capillary gripper than to a vacuum gripper. Moreover, the miniaturization will reinforce this advantage. Let us note that both capillary and vacuum grippers are well adapted to the picking of ﬂat components (with a low aspect ratio), which is not the case of the two-ﬁngered gripper, requiring opposite faces to grip the component. Another advantage of the capillary gripper is the contact damping due to the liquid ﬁlm (see later Fig. 11.5). At the contrary, the vacuum gripper can lead to cracks at the component surface, as indicated in Fig. 3.5. A ﬁnal argument concerning the good cycle time of the physical principle will be given in Chap. 12, showing that the performances of a capillary gripper can be assumed to be better than the typical picking time of the SMD assembly machines [174].

1

2

A Surface Mount Device is a component mounted on the surface of a printed circuit board. However, this limit is not clearly stated, a few tens of millimeters can be suggested as a lower limit.

5 Introduction

Beside van der Waals and electrostatic forces, surface tension eﬀects are often cited in the microassembly literature as being of utmost importance. The most frequently cited eﬀects are the inﬂuence on sticking during the release task [58], the negative eﬀect on the reliability of microswitches that can collapse by capillary forces [126], and the positive eﬀect of surface tension when chosen as gripping principle [70, 100]. This second part will consequently focus on the description of the underlying parameters of capillary and the modeling of the surface tension eﬀects in the simulation of the handling task. As the problematics of surface tension eﬀects is far from being easy, we will ﬁrst introduce a few parameters in Chap. 6 (those involved in the justiﬁcation of the capillary as a suitable gripping principle: surface tension, Young-Dupr´e equation, Laplace equation). After a review of the ways to take these surface eﬀects into account in handling (Chap. 7), we will present the static simulation at constant volume (Chap. 8). Two kinds of capillary force models, namely the energy based method and the Laplace equation based method, are compared and their equivalence is proven in Chap. 9. The application of these results to a watch ball bearing is presented in Chap. 10. We will then expound the advantages but also the limitations of this approach and introduce in Chap. 11 a second set of parameters and phenomena (static contact angle hysteresis, dynamic contact angle, inertial eﬀects of the meniscus, inﬂuence of surface roughness, and surface heterogeneities (impurities)). A study of the limits of the proposed method is presented in Chap. 12. Finally, models are developed to predict the separation distance between a gripper and a component that provokes their adhesion and to predict the separation distance causing the rupture of the liquid bridge linking them with each other (Chap. 13).

6 First Set of Parameters

6.1 Introduction The reader will ﬁnd in this chapter a brief introduction to the concepts of surface tension, contact angle, and Laplace equation, which will be used to model the liquid bridge and compute the capillary forces as it will be explained in Chap. 7. Based on these concepts, some arguments will be given at the end of the chapter to justify a priori the use of surface tension as a gripping principle.

6.2 Surface Tension A ﬁrst representation of the three states of matter is to consider that a solid is characterized by a volume and a shape and conversely, a gas does not have any own volume or own shape. A liquid is somewhere in between, having its own volume but no own shape. Usually, if a liquid is not contained, it spreads out. However, when we look at soap bubbles or small water droplets, we observe that they behave as if their surface was an elastic membrane, characterized by a “surface tension” that acts against their deformations. This surface tension is presented in a didactic way in [46]. The classical explanation of this phenomenon is based on the fact that in a liquid, the mutual attraction between the molecules overcomes the thermal agitation. All molecules inside the liquid are equally attracted by their neighboring molecules, but the molecules located at the interface between the liquid and (for example) a gas suﬀer from a so-called “attraction default” (see Fig. 6.1). From a thermodynamic point of view, the energetic state of a molecule near the surface is less favorable, leading to a global shape of the liquid that minimizes the interface area (this explains why wet hair stick together). This introduces the concept of surface energy (or surface tension), which has the dimensions of an energy by surface unit (J m−2 ). The mechanical point of view considers the surface tension as a tensile force by length unit (N m−1 ).

42

6 First Set of Parameters

Fig. 6.1. Illustration of the attraction default

The surface tension is denoted by γ and its numerical value depends on the molecular interactions: In most oils, the molecular interaction is van der Waals interaction, leading to quite low surface tensions (γ ≈ 20 mN m−1 ). As far as water is concerned, because of the hydrogen bonding, the molecular attraction is larger (γ ≈ 72 mN m−1 ). Typical values for conventional liquid range from 20 (silicone oil) to 72 mN m−1 (water at 20◦ C). For example, the following values for ethanol (23 mN m−1 ), acetone (24 mN m−1 ), and glycerol (63 mN m−1 ) are given in [46]. Surface tension is an important parameter in the perspective of a downscaling of the assembly equipment, because the force it generates linearly decreases with the size while the weight decreases more quickly. While surface tension has been pointed out as being one of the disturbing eﬀects in MEMS (stiction problems [104, 125, 184]), other uses have been positively considered [20, 81, 117, 55].

6.3 Young–Dupr´ e Equation and Static Contact Angle In the subsection 6.2 we saw that an interface between a vapor and a liquid could be characterized by an interfacial tension, denoted by γ and expressed as an energy by surface unit or as a force by length unit. Interfacial tensions can also be deﬁned at the interfaces between a liquid and a solid (γSL ) and between a solid and a vapor (γSV ). Typical values of γSV are given in [133]: Nylon (Polyamid) 6.6 (41.4 mN m−1 ), PE High density (30.3–35.1 mN m−1 ), PE Low density (32.1–33.2 mN m−1 ), PET (40.9–42.4 mN m−1 ), PMMA (44.9–45.8 mN m−1 ), PP (29.7), PTFE (20.0–21.8 mN m−1 ). The surface tension γ will indiﬀerently be denoted by γLV . When a droplet is posed on a solid substrate (see Fig. 6.2), the liquid spreads out and we can distinguish three phases (vapor, liquid, and solid) separated by three interfaces that join one another at the triple line, also called contact line. At this triple line, the liquid–vapor interface makes an angle θ with the substrate. If the contact line is at equilibrium, θ is called the static contact angle, which is linked to the interfacial tensions by the Young–Dupr´e equation [1, 89]: γLV cosθ + γSL = γSV .

(6.1)

6.4 Laplace Equation g

43

LV

Contact line liquid vapor

θ g

g

SL

SV

solid

Fig. 6.2. The Young–Dupr´e equation ∆A

gas liquid

θ solid

Fig. 6.3. Small displacement of the contact line

This equation can be written immediately by considering the balance of the forces acting on the contact line. A second approach is based on the fact that at the equilibrium the energy must be extremal and that any displacement of the contact line (see Fig. 6.3) leads to an energy variation equal to zero: ∆G = ∆A(γSL − γSV ) + ∆AγLV cosθ (6.2) lim∆A→∞ ∆G ∆A = 0, where ∆A and ∆G state for the variation of interface area and energy during the considered displacement. We will ﬁrst consider that this contact angle is constant as it is determined from the interfacial tensions that depend on materials in presence. This assumption will be used in the static simulation with constant volume. Measured values of the contact angles will be given in Sects. 17.9.5 and 19.6.

6.4 Laplace Equation Because of the surface tension, there exists a pressure diﬀerence across the interface between a liquid and a gas. In the case of a soap bubble for example, the pressure inside the bubble is bigger to compensate the outside pressure and to overcome the tension eﬀect. In a more general case, the pressure diﬀerence

44

6 First Set of Parameters dS

Pout

u v+dv

v

u+du

R1

R2 dq1

Pin

dq2

Fig. 6.4. Surface element of an interface between a liquid and a gas

is linked to the curvature of the interface according to the Laplace equation that will now be established. Let us assume a curved surface S at equilibrium on which we draw a net of coordinate curves u and v that intersect one another with an angle of 90◦ . Let us now consider a surface element dS limited by the curves u, v, u + du, and v + dv (Fig. 6.4). The diﬀerent forces that act on this surface element are (we only consider their components along the normal to the surface element, the positive direction being that of the external normal) the forces exerted by the internal and external pressures on dS (pin dS and pout dS), pin dS, pout dS

(6.3)

and the force exerted by the surface tension along the line v and v + dv (see Fig. 6.5) 2γdv sin

dθ1 , 2

(6.4)

where • •

factor 2 represents the fact that the surface tension acts along v and v + dv dv = R1 dθ2

A similar equation can be written as far as the surface tension along u and u + du is concerned: 2γdu sin

dθ2 . 2

(6.5)

We can now write the force balance along the normal to the surface element as

6.5 Eﬀects of a Liquid Bridge on the Adhesion Between Two Solids

45

n

du

γ

γ

R1

dq1

Fig. 6.5. Detail of the surface tension acting along v and v + dv

2γdu sin

dθ1 dθ2 + 2γdv sin + pout dS = pin dS. 2 2

(6.6)

Using the deﬁnition1 of dS = dudv, du = R2 dθ1 , dv = R1 dθ2 and using the classical approximation for small angles (sin x ≈ x), 6.6 can be rewritten as 1 1 + (6.7) = pin − pout . γ R1 R2 As (1/R1 + 1/R2 ) represents the double of the mean surface curvature H [47], 6.7 can be ﬁnally rewritten into the Laplace equation [1]: 2γH = pin − pout .

(6.8)

6.5 Eﬀects of a Liquid Bridge on the Adhesion Between Two Solids Let us now consider two solids linked by a liquid bridge2 , also called meniscus (Fig. 6.6). To link this to the general frame of micromanipulation, let us call the upper solid the “tool” or the “gripper” (it will be used as a gripper) and 1

2

Outside this section, θ1 and θ2 will denote the contact angles between the liquid and, respectively, the component and the gripper. The presented conﬁguration is axially symmetric, to introduce the capillary force from a “mechanical” point of view, i.e., using concepts like pressure or tensions. In a more general case, the conﬁguration is not axially symmetric and an energetic approach has to be implemented, see therefore Chap. 7.

46

6 First Set of Parameters z

Tool Gripper equationz2(r) pout

qs

r2

h

Interface q2

z

q1

pin Object

ρ

r'

Liquid bridge r1 r

Substrate

Fig. 6.6. Eﬀects of a liquid bridge linking two solid objects (Reprinted with permission from [108]. Copyright 2005 American Chemical Society.)

the lower one as the object (it will be used as micropart or microcomponent). Since axial symmetry is assumed, it can be seen in Fig. 6.6 that the contact line between the meniscus and the object (the gripper) is a circle with a radius r1 (r2 ). The pressure inside the meniscus is denoted by pin and that outside the meniscus by pout . θ1 is the contact angle between the object and the meniscus and θ2 is the angle between the gripper and the meniscus. z represents the separation distance (also called the gap) between the component and the gripper. h is called the immersion height. At its neck, the principal curvature radii are ρ (in a plane perpendicular to the z axis, i.e., parallel to the component) and ρ (in the plane rz). The object is submitted to the “Laplace” force, arising from the pressure diﬀerence pin − pout , and to the “tension” force, directly exerted by the surface tension. In what follows, we will consider that these two forces constitute what we will call the capillary force.3 The “Laplace force” is due to the Laplace pressure diﬀerence that acts over an area πr12 (see Fig. 6.7) and can be attractive or repulsive according to the sign of the pressure diﬀerence, i.e., according to the sign of the mean curvature: A concave meniscus will lead to an attractive force while a convex one will induce a repulsive force. FL = 2γHπr12 .

3

(6.9)

Marmur [124] uses the terms “capillary” force for the term arising from the pressure diﬀerence and “interfacial tension force” for that exerted by the surface tension.

6.6 A Priori Justiﬁcation of a Capillary Gripper

47

p in

Object

r1

pout

Fig. 6.7. Origin of the Laplace force: attractive case (Reprinted with permission from [108]. Copyright 2005 American Chemical Society.)

γ

q1 f1 Object

a

gz

q1

g SL

f1

g SV

a

Fig. 6.8. Origin of the tension force and detail (Reprinted with permission from [108]. Copyright 2005 American Chemical Society.)

The “tension force” implies the force directly exerted by the liquid on the solid surface. As illustrated in Fig. 6.8, the surface tension γ acting along the contact circle must be projected on the vertical direction, leading to FT = 2πr1 γ sin(θ1 + φ1 ).

(6.10)

Therefore, the capillary force is given by FC = FT + FL = 2πr1 γ sin(θ1 + φ1 ) + 2γHπr12 ,

(6.11)

φ1 denotes the slope of the component at the location of the contact line: it will be considered equal to zero in the following.

6.6 A Priori Justiﬁcation of a Capillary Gripper To roughly estimate the order of magnitude of the capillary and tension forces, let us assume that ρ ≈ r1 = 1 mm, ρ ≈ −10 µm and θ1 = 30◦ . Therefore, the

48

6 First Set of Parameters

meniscus has a mean curvature H = 12 (1/ρ +1/ρ) = −4.95×104 m−1 . If water is used, the surface tension γ is equal to 72 × 10−3 N m−1 . Consequently, we can expect that the gripping force Fgrip exerted by a water meniscus on a microcomponent is about Fgrip = FL + FT ≈ (−22.4 + 0.5) mN = −21.9 mN.

(6.12)

It is important to note that it is about the weight of a small cube with a 9.4 mm edge and 2, 700 kg m−3 density (aluminum). The ﬁrst temporary conclusion is that the surface tension eﬀects are large enough to pick up what we deﬁned as microcomponents. A second interesting element is the downscaling behavior of the capillary force. Most authors consider that the capillary force is a surface force. However, it must not be understood as being a force that depends on the second power of a characteristic length. This remark has already been pointed out in [19] and is well illustrated by the behavior of the capillary force that depends on the ﬁrst power of a characteristic length: indeed let us assume a conﬁguration (i.e., given γ, θ1 , θ2 , geometries) leading to curvature radii ρ and ρ and foot radius r1 (we assume φ1 = 0). Therefore, the force can be written as follows: 1 1 + (6.13) FC = 2πr1 γ sin θ1 + γ πr12 . ρ ρ If the characteristic size is reduced 10 times, 10 10 FC r1 r12 = . FC = 2π γ sin θ1 + γ + π 10 ρ ρ 100 10

(6.14)

The downscaling of this gripping principle is consequently very promising and at least more promising than the vacuum gripper whose force directly depends on the suction area (that is r12 ). A third advantage of the capillary gripper is that the squeeze ﬁlm eﬀect avoids damage to the handled part. We will discuss it again in subsection 11.4 (this damage at contact is often cited as a main drawback of the mechanical gripper). Moreover, like the vacuum gripper, the capillary gripper is particularly well suited to pick up plane objects or components with a very low aspect ratio. These microparts do not oﬀer enough space on their sides to be gripped by a mechanical gripper. Finally, the study and the modeling of the capillary as a gripping principle is a good way to enter microassembly. Capillary is involved on many scales, from the capillary condensation at nanotips to the reliability of microsystems. The use of capillary as a gripping principle is cited in [167] and has been tested in [70]. Grutzeck [70] set up a manipulation station using a gripper tip made of a silicon chip, with a small hole used to generate the gripping droplet. The gripped object was a silicon chip with a weight of 0.219 mN. In this application, the measured forces ranged from 11 to 21 mN with drop volumes between 1.6 and 0.19 µL (separation distance = 12 µm). The parameters of the object and the gripper are summarized below.

6.7 Conclusions

49

Table 6.1. Parameters of the manipulation proposed in [70] Element Gripper

Description

Value

Plane tip made of a silicon chip

4 × 4 mm2 , 600 µm height Φ400 µm

Droplet supply through a hole Object Plane tip made of a silicon chip Weight Surface properties of the chips Roughness Advancing contact angle (with water) Receding contact angle (with water) Advancing contact angle (with methanol) Receding contact angle (with methanol)

600 µm height 0.219 mN 1.36 µm 71◦ 50◦ 14◦ 13.5◦

6.7 Conclusions In this section, we presented some fundamental parameters involved in the capillary force (surface tension, contact angle, Laplace equation). According to 6.11, this capillary force can be calculated if the geometry of the liquid bridge is known but some approximations can be used in order to evaluate the force more quickly. This will be discussed in the next chapter.

7 State of the Art on the Capillary Force Models at Equilibrium

7.1 Introduction We will now compare several methods to estimate the capillary force in several conﬁgurations. Each method will be presented with one or two conﬁgurations but not all conﬁgurations will be detailed for all methods. The results are summarized at the end of this chapter. Most often the capillary forces are approximated by several formulations that include the following assumptions: (1) axial symmetry of the liquid bridge, (2) gravity eﬀects on the meniscus shape are neglected (in other words, the Bond number BO ≡ ∆ρgL2 /γ is assumed to be vanishing, which means meniscus with a size smaller than the capillary length LC ≡ γ/∆ρg - see later (8.1)). The main approaches are the following: • •

The energetic approach,1 consisting in deriving the interfacial energy W with respect to the separation distance between the gripper and the object Geometrical approximations of the meniscus shape – “Arc”, “circle”, or “toroidal” approximation – “Parabolic” approximation

7.2 Energetic Approach: Interaction Between Two Parallel Plates This method consists in the following: • 1

Writing the interfacial energy W of the system as a function of the parameters deﬁning the geometry of the system To be exact, the energetic approach requires the accurate knowledge of the meniscus shape, which can be computed by energy minimization (see for example the well known Surface Evolver software) or by numerically solving the Laplace equation in the axially symmetric case, see therefore Chap. 8).

52

7 State of the Art on the Capillary Force Models at Equilibrium z

S

r2

q2 z

L V q1

S

r1 r0

Fig. 7.1. Example of the energetic method: case of two parallel plates

•

•

Deriving this energy with respect to one of the parameters (the separation distance z is often used) to calculate the capillary force as a function of this parameter Estimating the derivative of the other parameters with respect to the chosen parameter by assuming a mathematical relation (for example, the conservation of the liquid volume)

This approach can be illustrated by the case of two parallel plates linked by a meniscus, as represented in Fig. 7.1. The system has three phases (S, solid; L, liquid; V, vapor) and three interfaces (LV, liquid–vapor; SL, solid–liquid; SV, solid–vapor) leading to a total energy equal to: W = WSL + WSV + WLV = γSL SSL + γSV SSV + γΣ,

(7.1)

where WSL = γSL1 πr12 + γSL2 πr22 ,

(7.2)

WSV = γSV1 (πr02 − πr12 ) + γSV2 (πr02 − πr22 ),

(7.3)

WLV = γΣ.

(7.4)

In these equations, r0 is an arbitrary constant radius, larger than the maximum between r1 and r2 and γSLi (γSVi ) state for the interfacial energy between solid i and the liquid (vapor). Σ states for the area of the liquid–vapor interface (the lateral area of the meniscus). As we try to get the expression of the force F acting on one of the plates along the vertical z as a function of the separation distance z, (7.1) must be derived with respect to z:

7.2 Energetic Approach: Interaction Between Two Parallel Plates

F =−

53

dW dz

dr1 dr2 dr1 − γSL2 2πr2 + γSV1 2πr1 dz dz dz dΣ dr2 + γSV2 2πr2 −γ . (7.5) dz dz To calculate all the derivatives involved in this expression, additional assumptions must be stated. The ﬁrst (and not restrictive) assumption is that the volume of the meniscus remains constant (we consequently do not consider the evaporation of the liquid). Unfortunately, this assumption is not suﬃcient and more restrictive assumptions must be added: = −γSL1 2πr1

1. The separation distance z is small compared to the radius r1 and r2 ; henceforth, we neglect the term depending on the lateral area Σ 2. The liquid volume can be approximated by V ≈ πr12 z.

(7.6)

Consequently, the conservation of the volume leads to dV dr1 = 2πr1 + πr12 = 0 dz dz

(7.7)

dr1 r1 =− . dz 2z

(7.8)

and

Moreover, let us make a third assumption: 3. The derivative of r2 with respect to z can be written in the same way, leading to r2 dr2 =− . dz 2z

(7.9)

With (7.8),(7.9), and (6.1), (7.5) can now be rewritten as F =−

πγ 2 (r cos θ1 + r22 cos θ2 ) z 1

(7.10)

or, in the case of two plates made of the same material F =−

2πγ 2 r cos θ. z

(7.11)

If we take the lateral area Σ of the meniscus into consideration, the previous assumptions must be replaced by these ones: 1. The meniscus shape can be approximated by a cylinder of radius r and height z

54

7 State of the Art on the Capillary Force Models at Equilibrium

2. The liquid volume is exactly given by V = πr2 z The lateral area Σ is given by Σ = 2πrz.

(7.12)

So, instead of neglecting it, we can write the derivative of Σ toward z as dΣ = 2πr. dz

(7.13)

Consequently, still considering the Young–Dupr´e equation (6.1), (7.11) can now be rewritten as F =−

2πγ 2 (r cos θ + rz). z

(7.14)

Or, if we write this latter equation as a function of the liquid volume V , 1

F =−

1

2γV cos θ 2γπ 2 V 2 − . 1 z2 z2

(7.15)

Finally, a last expression similar to (7.15) can be found in [60]: 1

lim F = −

z→0

1

2γV cos θ 2γπ 2 V 2 sin θ − . 1 z2 z2

(7.16)

To be comparable with (7.14) and (7.11), this last equation is rewritten to F =−

2πγ 2 (r cos θ + rz sin θ). z

(7.17)

We now propose several plots of (7.14), (7.11), and (7.17): 1. Parallel plates, inﬂuence of the contact line radius r for a contact angle equal to 15◦ (Figs. 7.2–7.5) 2. Parallel plates, inﬂuence of the contact angle for a contact line radius r of 100 µm (Figs. 7.6–7.9). It can be shown in these ﬁgures that the force given by (7.17) lies between the results of (7.14) and (7.11). Moreover, these three approximations tend to one another when z decreases. However, these equations are only approximations, and so it cannot be concluded that one would be more accurate than the other.

7.3 Energetic Approach: Other Conﬁgurations

55

R=1e−006m, θ=15°

0

10

0.01m

0.001m −5

Force [N]

10

0.0001m −10

1e−005m

10

1e−006m −15

10

10

−10

10

−8

−6

−4

10 10 Separation distance [m]

10

−2

Fig. 7.2. Comparison between (7.11), (7.14), and (7.17) for a contact line radius π , and a surface tension γ = 72 mN m−1 . Solid line: r = 1 µm, a contact angle θ = 12 (7.11), dashed line: (7.14), and dash–dot line: (7.17) R=1e−005m, θ=15°

0

10

0.01m

0.001m −5

Force [N]

10

0.0001m −10

1e−005m

10

1e−006m −15

10

10

−10

10

−8

−6

−4

10 10 Separation distance [m]

10

−2

Fig. 7.3. Comparison between (7.11), (7.14), and (7.17) for a contact line radius π , and a surface tension γ = 72 mN m−1 . Solid line: r = 10 µm, contact angle θ = 12 (7.11), dashed line: (7.14), and dash–dot line: (7.17)

7.3 Energetic Approach: Other Conﬁgurations Another important interaction is that between a sphere and a plane (see Fig. 7.10). It has been presented in [89] Chap. 15.6. (Details are also proposed in Appendix A.1.2). Let us consider a sphere with radius R, located at a distance z from a surface. The liquid meniscus rises up to a height z + h (the immersion height h is so that the ﬁlling angle φ is small). By considering a constant volume of liquid, Israelachvili [89] calculated that the attractive force between the sphere and the surface due to the liquid bridge is F =−

4πRγ cos θ 1 + hz

(7.18)

56

7 State of the Art on the Capillary Force Models at Equilibrium R=0.0001m, θ=15°

5

10

0

Force [N]

10

0.01m 0.001m

−5

10

0.0001m −10

1e−005m

10

1e−006m −15

10

10

−10

10

−8

−6

−4

10 10 Separation distance [m]

10

−2

Fig. 7.4. Comparison between (7.11), (7.14), and (7.17) for a contact line radius π and a surface tension γ = 72 mN m−1 . Solid r = 100 µm, a contact angle θ = 12 line: (7.11), dashed line: (7.14), and dash–dot line: (7.17) R=0.001m, θ=15°

5

10

0

Force [N]

10

0.01m 0.001m

−5

10

0.0001m −10

1e−005m

10

1e−006m −15

10

10

−10

10

−8

−6

−4

10 10 Separation distance [m]

10

−2

Fig. 7.5. Comparison between (7.11), (7.14), and (7.17) for a contact line radius π , and a surface tension γ = 72 mN m−1 . Solid line: r = 1 mm, a contact angle θ = 12 (7.11), dashed line: (7.14), and dash–dot line: (7.17)

and that maximum attraction occurs at z = 0, where F = −4πRγ cos θ.

(7.19)

More rigorous expressions valid for large φ and diﬀerent contact angles on each surface are given by [137]. Finally, similar developments lead to the force between two spheres with diﬀerent radii R1 and R2 and diﬀerent material (θ1 = θ2 ) (please refer to Appendix A.1.3 for further details): F = −4πγR cos θ,

(7.20)

where 2 cos θ ≡ cos θ1 + cos θ2 and 1/R ≡ 1/R1 + 1/R2 , so that it can be concluded that at contact and with small amounts of liquid (φ <<), the force

7.4 Geometrical Approach: Circle Approximation

57

R=0.0001m, θ=0°

5

10

0

Force [N]

10

0.01m 0.001m

−5

10

0.0001m −10

1e−005m

10

1e−006m −15

10

10

−10

10

−8

−6

−4

10 10 Separation distance [m]

10

−2

Fig. 7.6. Comparison between (7.11), (7.14), and (7.17) for a contact line radius r = 100 µm, a contact angle θ = 0, and a surface tension γ = 72 mN m−1 . Solid line: (7.11), dashed line: (7.14), and dash-dot line: (7.17) R=0.0001m, θ=15°

5

10

0

Force [N]

10

0.01m 0.001m

−5

10

0.0001m −10

1e−005m

10

1e−006m −15

10

10

−10

10

−8

−6

−4

10 10 Separation distance [m]

10

−2

Fig. 7.7. Comparison between (7.11), (7.14), and (7.17) for a contact line radius π , and a surface tension γ = 72 mN m−1 . Solid r = 100 µm, a contact angle θ = 12 line: (7.11), dashed line: (7.14), and dash-dot line: (7.17)

between two spheres with radii R1 and R2 is equal to that between a plane and a sphere of radius R given by 1/R = 1/R1 + 1/R2 .

7.4 Geometrical Approach: Circle Approximation Another widespread method in the literature to calculate the capillary force is to approximate the meniscus by an arc (part of a circle). This method requires the determination of ﬁve parameters (ro , zo , ρ, θmin , and θmax ) that are represented in Fig. 7.11. Several conditions can be expressed to determine these ﬁve parameters: the contact angles on each plate (therefore θmin = θ1 and θmax = π − θ2 ), the separation distance z. Another condition can be the volume of liquid or the mean

58

7 State of the Art on the Capillary Force Models at Equilibrium R=0.0001m, θ=30°

5

10

0

Force [N]

10

0.01m 0.001m

−5

10

0.0001m −10

1e−005m

10

1e−006m −15

10

10

−10

10

−8

−6

−4

10 10 Separation distance [m]

10

−2

Fig. 7.8. Comparison between (7.11), (7.14), and (7.17) for a contact line radius r = 100 µm, a contact angle θ = π/6, and a surface tension γ = 72 mN m−1 . Solid line: (7.11), dashed line: (7.14), and dash-dot line: (7.17) R=0.0001m, θ=80°

0

10

0.01m

0.001m −5

Force [N]

10

0.0001m −10

1e−005m

10

1e−006m −15

10

10

−10

10

−8

−6

−4

10 10 Separation distance [m]

10

−2

Fig. 7.9. Comparison between (7.11), (7.14), and (7.17) for a contact line radius r = 100 µm, a contact angle θ = 80◦ , and a surface tension γ = 72 mN m−1 . Solid line: (7.11), dashed line: (7.14), and dash-dot line: (7.17)

curvature of the meniscus. This last condition contains the largest approximation of this method: it assumes that the curvature will be constant but an axially symmetric meniscus with a circular proﬁle cannot have a constant curvature. The circle approximation method consequently consists in the following: • • •

Determining the circle parameters Computing its mean curvature (at the neck) Calculating the “Laplace” and “tension” components of the capillary force

Let us illustrate this method in the case of two parallel plates (see Fig. 7.11). Because of the geometry of the problem, we can express the circle radius ρ as a function of the separation distance z and the contact angles θ1 and θ2 :

7.4 Geometrical Approach: Circle Approximation

59

R

φ

h z

Fig. 7.10. Interaction between a sphere and a plane [89]: z is the gap between the sphere and the plane, h is the immersion height, Φ is the ﬁlling angle, and R is the sphere radius z

θ2 θmax z

ρ'

θ1

ρ

(r0 ,z0)

θmin r

r1

Fig. 7.11. Circle approximation of the meniscus: z is the gap between both plates, θ1 and θ2 are the contact angles, ρ and ρ are the two curvature radii at the neck

ρ=

z . cos θ1 + cos θ2

(7.21)

The coordinate z0 of the circle’s center is given by z0 = ρ cos θ1 =

z cos θ1 . cos θ1 + cos θ2

(7.22)

The coordinate ro can be expressed as a function of ρ and ρ : r0 = ρ + ρ .

(7.23)

We miss an additional condition in order to determine ρ : we will assume a constant volume V that can be expressed as V = πρ2 z.

(7.24)

60

7 State of the Art on the Capillary Force Models at Equilibrium

The arc is now completely determined by θmin = θ1 ,

(7.25)

θmax = π − θ2 , z , ρ= cos θ + cos θ2 1 V , ρ = πz r0 = ρ + ρ ,

(7.26) (7.27) (7.28) (7.29)

and we can compute the mean curvature H of this meniscus: 2H =

1 1 − . ρ ρ

(7.30)

ρ is positive but ρ must be counted negative because it contributes to the concavity of the meniscus. The diﬀerence of pressure across the interface is given by the Laplace equation (6.8) pin − pout = 2γH.

(7.31)

Both components of the capillary force can now be calculated as a function of r1 , the radius of the circular contact line on the plate 1, which is given by r1 = (ρ + ρ ) − ρ sin θ1 .

(7.32)

Henceforth, we deduce the “Laplace” force FL = πr12 (pin − pout )

(7.33)

and the “tension” force FT = 2πr1 γ sin θ1 ,

(7.34)

leading to a global capillary force equal to F = πr12 γ2H + 2πr1 γ sin θ1 (7.35) 1 1 − (7.36) = πγr12 + 2πγr1 sin θ1 ρ ρ cos θ1 + cos θ2 πz − = πγr12 (7.37) + 2πγr1 sin θ1 , V z where r1 = cos θ1 +z cos θ2 + V /πz − cos θ1 +z cos θ2 sin θ1 . Several authors used this method among whom Stifter et al. [161] and Marmur [124] apply the arc approximation to study the capillary force in surface force apparatus.

7.6 Comparisons and Summary

61

7.5 Geometrical Approach: Parabolic Approximation In some cases the arc approximation can lead to numerical diﬃculties. For example, Pepin et al. [142] emphasizes the switch from a convex meniscus to a concave meniscus: The center of the circle ﬁrst tends to −∞ before diverging to +∞. This diﬃculty can be avoided by using a parabolic approximation of the meniscus: r(z) = az 2 + bz + c.

(7.38)

The three unknown a, b, and c can be determined by imposing the contact angles θ1 and θ2 and the last unknown can be determined if either the pressure (i.e., the curvature: see also the Laplace equation) or the volume is imposed. For example, in the case of ﬂat components, the slope must match the contact angle θ1 : b=−

cos θ1 . sin θ1

(7.39)

The two other unknown can be determined by using an iterative scheme: 1. First choose a starting radius r 2. Compute the gripper point corresponding to r (i.e., determine z(r)) and the slope tan p of the gripper at this point 1 ( tan(θ12 + p) − b) 3. The parabola must match the contact angle θ2 ⇒ a = 2z 4. The point (r, z) belongs to the parabola too: c = r − az 2 − bz 5. Determine the corresponding volume v(r) 6. Compare v(r) and the prescribed volume V • If v(r) > V , decrease r • Otherwise, increase r With this geometrical shape, the switch from a convex to a concave meniscus smoothly occurs when a passes through zero before changing its sign.

7.6 Comparisons and Summary Comparisons between the diﬀerent formulations of the energetic method have already been proposed in the ad hoc section. In conclusion of this section devoted to the approximations, we propose the graphical comparison (Fig. 7.12) of the meniscus shapes between the geometrical methods. The impact on the pressure diﬀerence and the force is plotted in Fig. 7.13. As the diﬀerence between the “arc” and the “parabolic” approximations must be explained (the “better” approximation must be determined), we will turn ourselves to numerical solutions: According to the Laplace equation, we will compute the shape of a meniscus at equilibrium for a given volume. These simulations are of the highest interest as far as the use of capillary force as

62

7 State of the Art on the Capillary Force Models at Equilibrium −4

x 10

z [m]

6

4

2

0 2

4

6 r [m]

8

10

12 −4 x 10

Fig. 7.12. Comparison between the arc (solid line) and the parabolic (dashed line) approximations (Reprinted with permission from [108]. Copyright 2005 American Chemical Society). 500

8

0 Force [mN]

6 ∆p

−500 −1000 −1500

4 2

−2000 −2500 0

200

400 Gap [µm]

(a)

600

800

0 0

200

400 Gap [µm]

600

800

(b)

Fig. 7.13. Comparison between the meniscus shape models: arc (Open square) and parabola (Open triangle) models. (a) Pressure diﬀerence ∆p across the LV interface; (b) Force (Both reprinted with permission from [108]. Copyright 2005 American Chemical Society).

gripping principle is concerned. These simulations will consequently directly be related to the problematics of microgripping and microassembly presented in the state of the art of this work. Table 7.1 summarizes several classical approximations found in the literature and gives the corresponding references and assumptions (see Fig. 6.6 and Appendix A.1 for more details). The considered assumptions are as follows: 1. 2. 3. 4. 5.

Parallel plates Spherical tip (radius R) near a plate Arc approximation of the interface (where ρ = constant is the radius) Energetic formulation The radii r1 and r2 of the two circular contact lines are very small compared to R

7.6 Comparisons and Summary

63

Table 7.1. Summary of the capillary forces (sign “−” has been omitted) (Reprinted with permission from [111], Copyrights 2006 Koninklijke Brill N.V.) Ref. [124] [124] Eq.7.10 Eq.7.14 [89] [161] [71] [70] [48]

6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

Force F = 4πRγ cos θ (r1 /R)2 r1 /R F = 4πRγ cos θ{ 12 [ z/R+1−(1−(r 2 1/2 ] − [ 4 cos θ ]} 1 /R) ) πγ 2 2 F = z (r1 cos θ1 + r2 cos θ2 ) F = 2πγ r2 cos θ + 2πrγ z 4πRγ cos θ F = 1+(z/h) F = πγρ’2 ( ρ1 − ρ1 ) F = 2πγρ + γρ πρ2 z F = 2πγρ + γρ π 2 sin θ F = 2πr2 γ sin(θ2 + φ) + πr22 γ( ρ1 − r12 )

Assumptions 2,5,6,7,8,9 2,3,6,7,15 1,4,7,8,10,11 1,4,7,8,11,12 2,4,7,10,13,14 2,3,15 1,3,8,9 1,3,7,9 2,3

r1 = r2 = r (“symmetric case”) The contact angles are equal: θ1 = θ2 = θ The gap z is very small compared to the radius r of the contact line The curvature of the interface in the horizontal plane is negligible | ρ1 | << | ρ1 | Contribution of the interfacial energy liquid–vapor is neglected Constant volume V ≈ πr12 z Interfacial area liquid–vapor S ≈ 2πr1 z Constant volume V = Vcylinder − Vspherical cap , cf. Appendix A.1.2 Immersion height h is small (φ ≈ 0, cf. Fig. 7.10) Interfacial tension force is neglected

To end this section, let us add a recently published model [145] giving an analytical expression for the capillary force between two spheres with radii R1 and R2 as a function of the separation distance z: Fsphere/sphere = −

2R cos θ , 1 + z/(2h)

(7.40)

1 R2 where R is the equivalent radius given by R = R2R , 2 cos θ = cos θ1 + 1 +R2 cos θ2 , z is the separation distance or gap, and h is the immersion height, approximately given by [145]

h=

z (−1 + 1 + 2V /(Rz 2 )), 2

where V is the volume of the liquid bridge.

(7.41)

8 Static Simulation at Constant Volume of Liquid

8.1 Introduction This chapter details the computation of capillary forces based on the solving of the Laplace equation (6.8) in order to compute the meniscus shape. The equivalence between this method and the energy minimization approach will be treated in the next chapter.

8.2 Description of the Problem The problem set out in this section is the simulation of a handling task based on the capillary force as gripping principle. The object is pulled toward the gripper by the capillary force exerted by the liquid bridge. The simulation objectives will consist in determining this force and predicting the ability to perform the manipulation. As explained in the previous sections, the capillary force can be determined from the geometry of the system (i.e., liquid bridge, gripper, and component). In Fig. 8.2, we call “gripper” or “tool” the upper solid, characterized by a proﬁle described by the equation z = z2 (r). The lower solid is the handled component (“object”) whose proﬁle is described by the equation z = z1 (r). θ1 (θ2 ) is the contact angle between the liquid and the object (the gripper). We consider only axially symmetric menisci. The “manipulation” task can be split into three subtasks: the “gripping”(or “picking”) task, the “handling” task, and the “release” task. The gripper can be conical, spherical, parabolic, or cylindrical (other geometries can be added but have not been taken into account yet). They are characterized by two parameters: the gripper width and the gripper parameter p, deﬁned for each geometry in Fig. 8.1. The cylindrical gripper presents a horizontal plane tip and is actually a conical gripper with p = 0.

66

8 Static Simulation at Constant Volume of Liquid Gripper width p (radius)

Gripper width

Gripper width

p (aperture angle)

p (radius)

Fig. 8.1. Geometries of the gripper: conical, spherical, and parabolic gripper z

Tool Gripper equation z2(r) pout

qs

r2

h

Interface

q2 z

q1

pin Object

Substrate

ρ'

ρ Liquid bridge

r1 r

Fig. 8.2. Gripper, liquid bridge, and microcomponent (Reprinted with permission from [108]. Copyright 2005 American Chemical Society.)

8.3 Assumptions This simulation is based on several assumptions. Gravity eﬀects are neglected. Indeed, Charlaix [32] shows the existence of a cut-oﬀ length LC between capillary and gravity eﬀects, given by γ LC = . (8.1) ρg In the case of water (γ ≈ 72 × 10−3 J m−2 , ρ ≈ 103 kg m−3 , g ≈ 10 m s−2 ), LC ≈ 2.5 mm. This capillary length can also be interpreted from the ratio between the hydrostatic pressure (ρgL) and the Laplace pressure γ/L: γ 1 L2 ∆PLap = = C2 . 2 ∆PHyd ρg L L

(8.2)

8.4 Equations and Numerical Simulation

67

This result shows that the gravity eﬀects in the meniscus can be neglected if its characteristic length L is smaller than typically 1 mm. Let us note that this restriction applies to the meniscus and not to the solid parts (i.e., “gripper” and the handled part): to be more accurate, this restriction applies to the meniscus height. A second assumption is the static approximation: Liquid bridges are assumed to be at equilibrium. Moreover, the static contact angle hysteresis is neglected (i.e., the advancing contact angle is equal to the receding contact angle, see also p. 97). The validity of this assumption therefore depends on the liquid (for example, the diﬀerence between the advancing and the receding contact angles for methanol on Si chips is only 0.5◦ but can be quite larger, up to 20◦ in the case of water on the same substrate [70]). In the case of a 1 µL water droplet and a conical gripper (aperture angle of the cone is equal to 20◦ ) separated by a distance of 1 nm from a plane substrate, this assumption can lead to important variations of the capillary force. In the case of a contact angle θ2 = 40◦ , a contact angle θ1 of 50◦ (71◦ ) leads to a 922 µN (690 µN) capillary force (own results). However, in the picking task, the capillary force should be bigger than the weight. If we assume small vertical accelerations of the gripper, this implies that the gripped object will stay in close contact with the gripper and that the meniscus will be in a relative equilibrium position. Nevertheless, errors can occur in the simulation of the release task because the increasing separation distance between the object and the gripper implies the motion of the contact line. A third assumption is that not only the weight of the meniscus but also the inertial forces acting on the meniscus can be neglected during the simulation. This last assumption must be discussed in the case of a release task based on an important vertical acceleration communicated to the gripper, but is suitable in the simulation of the gripping task (see Chap. 12). Finally, we consider smooth surfaces for both solids and the evaporation of the meniscus is neglected (the volume of liquid is constant).

8.4 Equations and Numerical Simulation As presented in Chap. 6, the capillary force can be determined from the geometry of the liquid bridge, using (6.9) and (6.10). Therefore, the problem will consist in determining the shape of the meniscus from its diﬀerential formulation given by (6.8). This equation states that the mean curvature H of the meniscus is determined by (or determines) the pressure diﬀerence across the interface. If equilibrium is assumed and the liquid bridge height is smaller than LC , the pressure inside the liquid bridge will be constant. Therefore, H will be constant. To link the general formulation of equation (6.8) to the parametric description of a surface, let Σ be a surface given by its vectorial equation: Σ(u, v) ≡ OP = r(u, v),

(8.3)

68

8 Static Simulation at Constant Volume of Liquid

where O is a reference point and P a point of Σ, determined by the two parameters u and v. Let us assume the following diﬀerential operator [47]: ∂r ∂r . , ∂u ∂u ∂r ∂r . , F = ∂u ∂v ∂r ∂r . , G= ∂v ∂v 2 ∂ r L= .1n , ∂u2 ∂2r M = .1n , ∂u∂v ∂2r N = .1n . ∂v 2 The mean curvature H is given by (see Appendix B.2) 1 EN + GL − 2F M 1 1 , + = H= 2 R1 R2 2(EG − F 2 )

(8.4)

E=

(8.5) (8.6) (8.7) (8.8) (8.9)

(8.10)

where R1 and R2 represent the two principal curvature radii. For a zaxially symmetric meniscus, the surface equation is given by Σ(θ, z) ≡ r(θ, z) = r(z)¯ 1r + z ¯1z .

(8.11)

Using (8.10) and (8.11) the mean curvature of an axially symmetric surface is given by 2H =

r 1 − , (1 + r2 )3/2 r(1 + r2 )1/2

(8.12)

where r = ∂r/∂z and r = ∂ 2 r/∂z 2 . Since the orientation of ¯1n is not oneto-one, the sign of H must be determined. In Fig. 8.3, we see a spherical soap z r''<0

r>0

z(r)

r Fig. 8.3. Case of a spherical bubble with pin > pout

8.4 Equations and Numerical Simulation

69

z rP

fP

P

q2

zP

q1

r

Fig. 8.4. Boundary conditions depend on the geometry and materials

bubble in air with an internal pressure higher than the external one, allowing to write <0

r ∆p 1 pin − pout − ≡ . + = 2 1/2 γ γ (1 + r2 )3/2 r (1 + r ) >0

>0

>0

(8.13)

>0

Let us put (8.13) as a system of two ﬁrst-order diﬀerential equations:

dr u = dz du dz

=

1+u2 r

−

∆p γ (1

+ u2 )3/2 .

(8.14)

To solve these equations, ∆p must be known and boundary conditions have to be set (Fig. 8.4): let us assume that we know the point P of the meniscus in contact with the gripper. Therefore, zP and rP = r(zP ) are given by the initial coordinates of P and the slope of the meniscus in P is given by 1

uP =

, dr |zP = { tan(φP +θ2 ) dz 0,

if (θ2 + φP ) = if (θ2 + φP ) =

π 2 π 2

,

(8.15)

where the contact angles θ2 and φP depend on the gripper geometry. In our problem (how to determine the meniscus for given contact angles θ1 and θ2 and liquid volume V ), only θ2 is known. Indeed, ∆p and the position of P are a priori unknown. Ref. [48] already suggested to iterate on ∆p to adjust θ1 to the prescribed value. The typical evolution of θ1 as a function of ∆p in our problem in shown in Fig. 8.5: Increasing pressure diﬀerence (i.e., more negative ∆p) leads to a more curved meniscus, and consequently, to smaller θ1 . Details will be put forward in Chap. 14. P is still unknown and the condition on V has not yet been used. Therefore, a second iteration loop is used [110] to determine P : An initial position of P is guessed in order to solve the ﬁrst iteration loop

70

8 Static Simulation at Constant Volume of Liquid 100 90 80 70

θ1

60 50 40 30 20 10 0 −300

−250

−200

−150 dp

−100

−50

0

Fig. 8.5. Contact angle θ1 (◦ ) as a function of the pressure diﬀerence across the interface dp (Pa) x 105

z (m)

8 6

Gripper Starting point i+1 Starting point i

Meniscus

Starting point 1

4 2 0 0.5

1

1.5 r (m)

Meniscus ij (starting point i, ∆ pj)

2 x 10−4

Fig. 8.6. The double iterative scheme for a spherical gripper (R = 0.1 mm), water, θ1 = θ2 = 30◦ , V = 4.5 nL, z = 0. Meniscus ij is obtained with the ith starting point and the jth pressure diﬀerence (Reprinted with permission from [108]. Copyright 2005 American Chemical Society.)

(i.e., determine a meniscus that would be correct as far as contact angles are concerned), leading to a candidate whose volume is computed. If this volume is smaller (larger) than the prescribed one, P is moved away (closer) from the symmetry axis (this is achieved by dichotomous search). In Fig. 8.6, P is successively deﬁned by the following radii: 50, 75, 87.5, 93.8, 96.9, and 98.4 µm. This double iterative scheme is illustrated in Fig. 8.7: it is actually an application of the so-called shooting method.

8.5 Discussion and Conclusions

71

Choose a meniscus starting point Choose a pressure difference Contact angle OK?

No

Volume OK?

No

Compute the capillary force

Fig. 8.7. View of the resolution (Reprinted with permission from [108]. Copyright 2005 American Chemical Society.) θ1 θ2 γ

Simulation

Geometries

Force

V z

Fig. 8.8. Inputs and outputs of the static simulation

8.5 Discussion and Conclusions The simulation described in this section can be qualiﬁed as a static simulation at constant volume. Its main characteristics are the following: • • • •

Actual geometries of the gripper and the object can be taken into account Material properties are taken into consideration with the contact angles, the surface tension Liquid volume is a user parameter Capillary force is computed

The main assumptions of the model are as follows: • • • • •

Gravity is neglected toward the capillary force Contact angle hysteresis is not taken into consideration Dynamic contact angle is not taken into account Materials are assumed to be smooth (no roughness) and without any impurities Inertial forces acting on the liquid bridges are neglected, which is not valid if the release task is based on acceleration of the gripper

72

8 Static Simulation at Constant Volume of Liquid

Not all these assumptions are restrictive, and we will consider additional models in Chap. 11 that could be used to extend the validity of this simulation to more complex conﬁgurations involving surface roughness or surface impurities (i.e., how to modify θ1 and θ2 in Fig. 8.8). Chapter 9 will establish on a formal way the equivalence between the energetic approach presented in Chap. 7 and the approach based on the sum of the “tension” term and the “Laplace” term as illustrated in the current chapter. Obviously, the reader can also go directly to Chap. 10, dealing with the application of these models to a watch ball bearing case study.

9 Comparisons Between the Capillary Force Models

9.1 Introduction This chapter gives evidence of the equivalence between the energetic approach and the direct formulation based on the Laplace and the tension terms: F = FL + F T = −

dW , dz

(9.1)

where FL is given by (6.9), FT by (6.10) and W by (7.1). z is the separation distance between both solids.

9.2 Qualitative Arguments The energetic approach involves both Laplace and tension terms. Let us illustrate this in the case of two parallel plates (see Fig. 7.1 reprinted as Fig. 9.1) separated by a distance z (for convenience, both contact angles have been chosen to be equal to θ). Based on the arguments of Sect. 7.2, the force was given by (7.15) F =−

2γV cos θ 2γπ 1/2 V 1/2 − . 1 z2 z2

(9.2)

Let us now consider the case θ = π/2 to compare the force derived from the energy with that from the meniscus geometry. Indeed, in this case, the approximation of the meniscus shape by a cylindrical volume is exact (stripped lateral area in Fig. 9.1), and we can directly compute the mean curvature of this cylinder. Consequently, we have to compare the force derived from the energy: πV (9.3) F = −γ z

74

9 Comparisons Between the Capillary Force Models z

S

z

r2

L

q2 V

q1

S

r1 r0

Fig. 9.1. Case of two parallel plates separated by a gap z

and the one established from the mean curvature, which is given by: 2H =

1 + 0, r

(9.4)

leading to a pressure diﬀerence ∆p = 2Hγ =

γ r

(9.5)

and henceforth to a “Laplace” term of the force equal to FL = πr2 ∆p = πγr

(9.6)

Note that this term is positive, i.e., repulsive, because the meniscus is convex, leading to a positive pressure diﬀerence. The ‘tension’ term of the force FT can be written as FT = −2πrγ,

(9.7)

leading to a total capillary force equal to F = FL + FT = −γπr.

(9.8)

Assuming a cylindrical shape for the meniscus (V = πr2 z as already stated in (7.6)), the latter equation can be rewritten into πV . (9.9) F = FL + FT = −γ z Since (9.3) and (9.9) are equal, we conclude that the force derived from the energy exactly represents both the terms of the capillary force (note well that the expression F = 4πγR cos θ proposed at (15.35) of [89] has been derived this way, consequently including both terms).

9.3 Analytical Arguments

75

z

R

Fig. 9.2. A sphere (radius R) and a plate separated by a gap z: both contact angles are equal to θ = π/2

Another argument is geometric. Let us consider the case depicted in Fig. 9.2 where both contact angle are equal to π/2. The idea behind the following intuitive argumentation is to prove that the energetic method well involves both terms. On the one hand, the case depicted in Fig. 9.2 can be modeled, thanks to (7.18) where h is set equal to 0: F = 4πRγ cos θ. As indicated in Sect. A.1.3, this expression is based on the energetic approach. In the proposed case, this equation leads to F = 0. On the other hand, the meniscus is clearly convex, leading to a repulsive “Laplace” force which is here counted strictly positive. Now, if we take the (always) attractive “tension,” we see (qualitatively) that we could have a total force equal to zero. This argument also conveys the idea that both approaches are equivalent.

9.3 Analytical Arguments 9.3.1 Deﬁnition of the Case Study We propose to demonstrate the equivalence of the approaches on a prism– plane conﬁguration. The prism is deﬁned by its length in the y direction, L, and its angular aperture φ (see Fig. 9.3). Its location is deﬁned by the distance1 D between its apex A and the plane. Let us assume a volume of liquid V wetting the plane with a contact angle θ1 and the prism with a contact angle θ2 . Since the curvature of the meniscus in the direction y perpendicular to 0xz is equal to zero, the Laplace equation (8.13) becomes x ∆p , = γ (1 + x2 )3/2

(9.10)

where x = dx/dz. 1

For the sake of clarity, since z will be used as one of the coordinates, the gap is noted D in this chapter.

76

9 Comparisons Between the Capillary Force Models z α θ2

x2

φ ρ

θ2

h A

D O

C

l

φ

ρ

θ1

zo x

x1 xo

Fig. 9.3. Prism–plane conﬁguration

Assuming a vanishing Bond number, the hydrostatic pressure inside the meniscus is neglected by comparison to the Laplace pressure diﬀerence ∆p, which is therefore constant in all the meniscus. Therefore, the right hand side of (9.10) is constant and this equation can be integrated twice with respect to z to ﬁnd the relation x = x(z), with two integration constants and the undeﬁned pressure diﬀerence ∆p. A more straightforward derivation is based on the fact that since one of the curvature radii is inﬁnite and that the total curvature 2H is a constant, the second curvature radius (1 + x2 )3/2 /x is a constant: let us denote it ρ. Therefore, the meniscus proﬁle is a curve with constant curvature, i.e., a circle given by the equation (x − x0 )2 + (z − z0 )2 = ρ2 ,

(9.11)

where x0 and z0 are the coordinates of the circle center. Once again, three parameters are to be determined: x0 , z0 , and ρ. This can be done using three boundary conditions: both contact angles θ1 and θ2 and the volume of liquid V . 9.3.2 Preliminary Computations Let us express x0 , z0 and ρ as functions of known data (φ, D, θ1 , θ2 ) and the immersion height h, which is still unknown at this step, but which will be determined using the condition on the volume of liquid V . Note that x2 is an intermediary variable and that x1 will be used later. For sake of convenience, the notation α = θ2 + φ has been adopted in the following equations:

9.3 Analytical Arguments

77

h , tan φ D+h ρ= , cos θ1 + cos α z0 = ρ cos θ1 , x0 = x2 − (z0 − D − h) tan α,

(9.14) (9.15)

x1 = x0 − z0 tan θ1 .

(9.16)

Additional useful relations are the meniscus equation: x = x0 − ρ2 − (z − z0 )2 ,

(9.17)

(9.12)

x2 =

(9.13)

the meniscus slope x : x = −

z − z0 , x − x0

(9.18)

and ﬁnally, the rewritten Laplace equation linking ∆p and ρ: ∆p =

γ . ρ

(9.19)

h is still to be determined using the volume of liquid V (see next step). 9.3.3 Determination of the Immersion Height h The volume of liquid can be used to determine the value of the immersion height h, starting from the following expression of V as illustrated in Fig. 9.4: V = 2LA = 2L[x0 (h + D) − AI − AII − AIII − AIV ],

(9.20) (9.21)

z

x2

D O

AII

AI

h

C

AIV A

AIII

zo x

x1 xo

Fig. 9.4. Determination of the immersion height from the volume of liquid

78

9 Comparisons Between the Capillary Force Models

where x2 h , 2 (x0 − x2 )(D + h − z0 ) , AII = 2 z0 (x0 − x1 ) , AIII = 2 ρ2 (π − α − θ1 ) AIV = . 2 Therefore, the equation giving the volume V can be rewritten as x2 h V = 2L x0 (D + h) − 2 2 ρ (π − α − θ1 ) (x0 − x2 )(D + h − z0 ) z0 (x0 − x1 ) − − − 2 2 2

AI =

(9.22) (9.23) (9.24) (9.25)

(9.26)

= L 2x2 D + x2 h +ρ [sin α cos α + 2 sin α cos θ1 − π + α + θ1 − sin θ1 cos θ1 ] 2

= L h2

(9.27)

≡µ(cos θ1 +cos α)2

1 1 2 + µ + 2hD + µ + µD tan φ tan φ

(9.28)

This latter equation can be rewritten as a second-degree equation with respect to the unknown h: h2 + 2hD + which leads to h = −D ±

µD2 − V /L = 0, µ + tan1 φ

D2 −

D2 µ − V /L . µ + tan1 φ

(9.29)

(9.30)

The “−” solution makes no physical sense since the immersion height cannot be negative. Consequently D2 µ − V /L h = −D + D2 − (9.31) µ + tan1 φ and the variation of h with respect to a variation of the separation distance D (it will be used in what follows) is given by dh D 1 = −1 + . dD D + h 1 + µ tan φ

(9.32)

9.3 Analytical Arguments

79

9.3.4 Laplace Equation Based Formulation of the Capillary Force As it has been previously explained, the capillary force can be written as the sum of a term depending on the Laplace pressure diﬀerence ∆p and the so-called tension term: F = 2Lx1 ∆p + 2Lγ sin θ1 x1 = 2Lγ + sin θ1 ρ x0 = 2Lγ ρ x2 D + h − z0 = 2Lγ + tan α ρ ρ h cos θ1 + cos α + sin α . = 2Lγ D+h tan φ

(9.33) (9.34) (9.35) (9.36) (9.37)

Using (9.31) the force can be expressed as a function of the volume of liquid V , the separation distance D, and the angles of the problem: contact angles θ1 and θ2 on the one hand and the prism angle φ on the other hand. Let us remind that α = θ2 + φ. 9.3.5 Energetic Formulation of the Capillary Force As previously explained, the energetic or thermodynamic approach is based on the diﬀerentiation of the total surface energy W with respect to the separation distance D: W = γ(Σ − A1 cos θ1 − A2 cos θ2 ),

(9.38)

where Σ = 2L = 2Lρ(π − α − θ1 ), A1 = 2Lx1 , h . A2 = 2L sin φ Consequently, the reduced surface energy W/(2Lγ) can be written as arbitrary constant) W h cos θ2 = ρ(π − α − θ1 ) − cos θ1 + ρ sin α − ρ sin θ1 − h 2Lγ tan φ sin φ π − α − θ1 − sin α cos θ1 + sin θ1 cos θ1 = (D + h) cos θ1 + cos α

(9.41) (+ any

(9.42)

≡β

cos θ1 cos θ2 + tan φ sin φ cos θ2 cos θ1 − . = Dβ + h β − tan φ sin φ −h

(9.39) (9.40)

(9.43) (9.44)

80

9 Comparisons Between the Capillary Force Models

To compute the force from the energy, the latter equation has to be derived with respect to D using (9.32) (β is constant with respect to D): dW 1 D 1 cos θ2 cos θ1 = β + −1 + − β− (9.45) dD 2Lγ D + h 1 + µ tan φ tan φ sin φ cos θ2 cos θ1 + = tan φ sin φ D 1 cos θ2 cos θ1 + − β− , (9.46) D + h 1 + µ tan φ tan φ sin φ where µ, h, and β have been deﬁned in (9.27), (9.31), and (9.43). All the other parameters are given data. It should now be proved that (9.37) and (9.46) are equivalent. 9.3.6 Equivalence of Both Formulations Equation (9.46) can be rewritten as

dW cos θ1 cos θ2 D π − α − θ1 − sin α cos θ1 + sin θ1 cos θ1 = + + 2LγdD tan φ sin φ D+h (cos θ1 + cos α)(1 + µ tan φ) cos θ1 cos θ2 1 + − (9.47) 1 + µ tan φ tan φ sin φ

It is shown in Appendix C that the expression in brackets in (9.47) is equal to −(cos θ1 + cos α)/ tan φ. Therefore, (9.47) can be rewritten into dW 1 cos θ1 cos θ2 D cos φ = + − (cos θ1 + cos α) dD 2Lγ tan φ sin φ D+h sin φ (cos θ1 cos φ + cos θ1 )(D + h) − D cos φ(cos θ1 + cos θ2 cos φ − sin θ2 sin φ) . = (D + h) sin φ (9.48)

To let appear the term sin α present in (9.37), let us add and substract sin α simultaneously to the latter equation: after some (tedious) calculations and using the relation α = θ2 + φ, the following expression can be obtained dW 1 h cos φ(cos θ1 + cos θ2 cos φ − sin θ2 sin φ) = + sin α dD 2Lγ (D + h) sin φ h cos θ1 + cos α = + sin α. D+h tan φ As a conclusion, the latter equation leads to a force given by h cos θ1 + cos α dW F =− = −2Lγ + sin α . dD D+h tan φ

(9.49)

(9.50)

The negative sign in front of 2L indicates that the force is attractive. Consequently, it is concluded that the force computation based on the Laplace equation (9.37) and the expression obtained from the energy formulation (9.50) are equal.

9.4 Conclusions

81

9.4 Conclusions It is shown that both approaches are equivalent; it means the energetic approach already involves the tension term and the Laplace term on an implicit way. Consequently, the energetic approach as proposed by Israelachvili (see (7.18)) includes both terms, even if, for zero separation distance, the pressure term usually dominates the tension one. For axially symmetric conﬁgurations, the method based on the Laplace equation will be preferred because it can be easily numerically solved.

10 Example 1: Application to the Modeling of a Microgripper for Watch Bearings

10.1 Introduction This chapter aims at applying the force models presented in the previous chapters to the case study of a watch ball bearing.

10.2 Presentation of the Case Study The proposed case study consists in the design of a capillary gripper to assemble the balls of a watch bearing (Fig. 10.1a), described in [139]. It is made of the following components: • • • • •

A stainless steel (4C27A) inner ring1 A stainless steel (4C27A) outer ring2 An austenitic steel (AISI 301) cage3 Zirconium oxide balls, with diameters 300 and 500 µm A stainless steel (4C27A) cone4

The assembly task can be sketched by the insertion of a ball in a hole, such as displayed in Fig. 10.1b. One of the requirements is to avoid the conventional tweezers and vacuum grippers, because of the scratches they provoke on the balls (Fig. 3.5). Due to the very small weight of the balls (about 3.8 µN), the surface tension based gripper is largely strong enough since it generates forces up to 150 µN. The handling scheme is illustrated in Fig. 10.1b: The picking force is provided by the capillary force and the releasing task is ensured by laterally moving the gripper once the ball is in the hole. Since the gripper uses capillary forces, a liquid has to be dispensed before each manipulation, 1 2 3 4

In In In In

French: French: French: French:

le noyau. la bague. la cage. le cˆ one.

84

10 Example 1: Application to the Modeling of a Microgripper

(a)

(b)

Fig. 10.1. (a) Ball bearing view (courtesy of MPS – http://www.mpsag.com/). (b) Handling scheme of the capillary gripping for the insertion of a ball in a hole (Reprinted with permission from [112]. Copyright 2006 Institute of Physics.) Ø0.1 Liquid channel M 1.7

Reference surface

Gripper tip 20˚

Ø0.5

(a)

(b)

(c)

Fig. 10.2. Designed capillary gripper. (a) Schematic view: the so-called reference surface is the surface which contacts the ball during the handling; (b) prototype (the distance between the graduations is 1 mm); (c) tip detail (the conical surface inclined to an angle 20◦ can be seen) (Reprinted with permission from [112]. Copyright 2006 Institute of Physics.)

but there is no need to eject this liquid (such as for example in ink-jet printing): it is suﬃcient to bring a bit of liquid in contact with the ball to pick up (see further). Besides this dispensing functionality, the other functions of the developed gripper tip can be summarized as follows: (1) to develop a picking force larger than the weight of the object (W ≈ 3.8 µN); (2) to develop a picking force large enough to handle the component with reasonable accelerations (manual handling); (3) to center the ball with respect to the gripper to ensure its positioning; (4) to release the ball once it is inserted in the hole. The proposed gripper is shown in Fig. 10.2b and c. Prototypes have been machined in stainless steel and some of them have been coated with a hydrophobic silane-based coating. Two solutions have been tried to supply the gripping liquid: (1) to drive the pressure through the gripper channel (see Fig. 10.2a); (2) to dip the tip in the liquid. Finally, the ﬁrst solution has been discarded because of the

10.2 Presentation of the Case Study

G

Response time

F

Force

85

Viscosity

Liquid

Surface tension

Roughness Material Impurities

B A

Contact angle

Solid

Channel C

Geometry

Gripper geometry

Volume of liquid

D

Hanging volume

E

Actual volume

Fig. 10.3. Overview of the required models

instabilities of the generated droplet when its height approaches half the diameter of the gripper. It has been taken advantage of the good repeatability of the volume transferred to the gripper by dipping it into the liquid (and consequently, the internal channel has been suppressed). The study of this surface tension gripper involves several models: the network linking the parameters is shown in Fig. 10.3. The solid considered in this problem is related to its wetting properties, determined at least by the type of material, the surface roughness, and the surface impurities, i.e., the chemical heterogeneousness of the surface. These parameters therefore deﬁne the kind of solid through model A, which is unknown. This solid surface, together with the surface tension of the liquid, determines the contact angle, through model B. What we call model B covers for example the Cassie or Wenzel models which take the surface roughness into account (see Sect. 11.3). Unfortunately, these ones are not suﬃcient to quantitatively describe all the information of contact angles: the hysteresis for example cannot be predicted. Therefore, model B is also considered as unknown, and the contact angles will be measured as inputs of the models F and G (see Sect. 19.6 for the contact angles measurement). The grippers are essentially deﬁned by their so-called geometry (i.e., the diameter of their tip, which is equal to the ball diameter, the value of the angle of their conical tip (20◦ in Fig. 10.2a)), and for some of the grippers the presence of an internal channel initially intended for the liquid supply. The characterization of these geometries (model C) is summarized in Sect. 19.2. The proposed tip dipping method is labeled D in Fig. 10.3 because it determines the volume of liquid hanging below the gripper tip. Unfortunately, at the ﬁrst contact between the gripper and the component, a part of this

86

10 Example 1: Application to the Modeling of a Microgripper

liquid is left on the component and therefore the hanging volume of liquid is diﬀerent from the actual volume of liquid involved in handling the task. Unfortunately, the relation between both volumes (model E) is neither easy to described nor easy to measure. In a static description, the force developed by the gripper depends on the surface tension, the contact angles, the gripper, geometry and the actual volume of liquid. This relation is called model F and can be described either analytically (see Sect. 10.3) or numerically (see Sect. 10.4). Note that both models have been benchmarked (see Sect. 10.5) and that an experimental validation is partially presented in Sect. 20.5.

10.3 Analytical Model Based on the Circle Approximation The theoretical capillary force developed by the gripper can be computed by using the so-called circle approximation presented in Sect. 7.4 (the detailed problem is shown in Fig. 10.4). Let us consider the situation depicted in Fig. 10.4b: the meniscus wets the ball along the circle containing B. (The

D

d Gripper bearing surface

J F

E

θ1

I

θ2

x H

α

B R

R2

β

A

z

I

G

C

α

C

β

RB

O B

θ

Rsinb

R1 R Handled ball

Symmetry axis

(a)

β A

(b)

Fig. 10.4. (a) The conﬁguration to model, including a cylindrical approached meniscus shape; (b) detailed view of the meniscus and the contact angles (Reprinted with permission from [112]. Copyright 2006 Institute of Physics.)

10.3 Analytical Model Based on the Circle Approximation

87

symmetry axis of Fig. 10.4b is perpendicular to this circle and contains its center. The position of this circle, whose radius is equal to RB , is determined by the ﬁlling angle β.) The contact angle θ1 is the angle between the tangent to the meniscus on the one hand and the tangent to the ball on the other hand: It is determined by the wetting properties of the materials, i.e., by the triple {handling liquid, material of the ball, surrounding environment}. On the gripper side (point C), since the gripper and the ball can be made of diﬀerent materials, the contact angle θ2 can be diﬀerent from θ1 . The circle approximation assumes that the meniscus has a circular shape, centered in O and with a radius R2 . Note that in Fig. 10.4a, D has been chosen two times the ball radius; indeed, the gripper has been designed as large as possible (for manufacturing reasons and to get a larger capillary force) without being larger than the ball to allow it to enter inside the bearing. With these assumptions, the capillary force5 exerted by the meniscus on both solids can be computed according to (6.11) and using 2H =

1 1 + , R1 R2

(10.1)

where R1 and R2 are the principal curvature radii of the meniscus. R2 can be directly read in Fig. 10.4b and, according to its deﬁnition, R1 is given by R1 = Since θ = ten as R1 =

RB . cos θ π 2

(10.2)

− θ1 − β and RB = R sin β, the curvature radius R1 can be writ-

R sin β , sin (θ1 + β)

(10.3)

where R is the ball radius, β is the ﬁlling angle (which is a function of the dispensed volume of liquid as it will be discussed later), and θ1 is the contact angle at the ball side. R2 can be determined by writing the distance z (Fig. 10.4b) on two different ways: z = R(1 − cos(β − α)) = R2 (cos θ2 + cos(θ1 + β − α)),

(10.4)

where α is given by the geometry of the gripper (in Fig. 10.2a, α = 20◦ ). Consequently, the capillary force of (6.11) can be rewritten into sin(β + θ1 ) cos θ2 + cos(θ1 + β − α) 2 + F = πR sin βγ sin β 1 − cos(β − α) (10.5) +2πR sin βγ sin(β + θ1 ) 5

Since only equilibrium situation is considered, and since the meniscus weight can be neglected – due to a vanishing Bond number’ – the forces exerted by the ball and the gripper onto the meniscus balance. Therefore, the (reaction) forces exerted by the meniscus on the ball and on the gripper are equal.

88

10 Example 1: Application to the Modeling of a Microgripper

This equation gives the force as a function of the handling liquid (γ), the materials (θ1 , θ2 ), the size of the ball (radius R), the volume of liquid through the ﬁlling angle β, and the gripper geometry (α). Let us note that the meniscus is not deﬁned for β = α, leading to nonphysical result for the force (the force tends toward inﬁnity, which is discussed again in Sect. 10.6). The link between the dispensed volume of liquid V and the ﬁlling angle β can be determined as follows (see also the result in Fig. 10.5): In Fig. 10.4a, the dispensed volume of liquid ﬁlls the BCEF GH area between the ball and the gripper, therefore V = V1 − V 2 − V 3 ,

(10.6)

where V1 is the sum of the volumes of the cylinder BCF G and the cone CEF , V2 is the volume of the spherical cap of the ball limited by the ﬁlling angle β, and V3 is the volume EJF . These volumes are given by the following equations: π πR3 sin β 2 (1 − cos(β − α)) + R3 sin β 3 tan α, cos α 3 1 2πR3 3 V2 = 1 − cos β + cos3 β , 3 2 2 3 d V3 = π tan α. 24 V1 =

(10.7) (10.8) (10.9)

The results of this analytical model are given in Sect. 10.5, together with those of the numerical model presented in Sect. 10.4. 60

Filling angle [˚]

50 40 30 20 10 0 0

0.5

1 Volume of liquid [m³]

1.5

2 x 10−12

Fig. 10.5. Filling angle β in degree as a function of the volume of liquid in m3

10.4 Numerical Model Based on the Laplace Equation

89

10.4 Numerical Model Based on the Laplace Equation The method presented in Sect. 8.4 has been adapted to the current case study, that is the two ﬁrst-order diﬀerential equation systems (8.14) can be rewritten into Y = f (z, Y ), where

(10.10)

r Y = , r

f (z, Y ) =

1+Y (2,1)2 Y (1,1)

Y (2, 1) ∆p − γ (1 + Y (2, 1)2 )3/2

.

As already mentioned, this equation can be solved only if ∆p is known (and actually it is not in advance) and with the following initial conditions for z0 = R cos β (see Fig. 10.6): R sin β . (10.11) Y0 = tan β + θ1 − π2 Since ∆p is not known in advance, it will be guessed and iterated until the computed meniscus converges to the angle θ2 on the gripper side. The algorithm can be expressed as follows: 1. Choose β 2. Choose ∆p z z + r tanα - R cosα - R sinαtanα = 0

z

z + r tanα - R cosα- R sinα tanα = 0 θ1

90−b

θ

α

B

r = r(z)

α

B z0

Event detection

z0

A

A

R

R

β

β

z - r cotanb = 0 r

r

r0

r0

Symmetry axis

Symmetry axis

(a) Fig. 10.6. (a) Initial conditions: tan θ = interval for the pressure diﬀerence

(b) dr dz

; (b) rough evaluation of the initial z=0

90

10 Example 1: Application to the Modeling of a Microgripper

We propose to evaluate an approximated initial interval for ∆p, so that a dichotomic search can be achieved (the nonanalytical function between θ2 and ∆p is monotonic). To do so, let us compute the distance L between A and B in Fig. 10.6, given by L2 = (rA − rB )2 + (zA − zB )2 2 R tan β − rB = R sin β − cos(α)(1 + tan α tan β) 2 R + cos β − . cos α(1 + tan α tan β)

(10.12)

From this distance, we deduce the radii ρ1 and ρ2 of curvature of the extremal approached menisci shown in Fig. 10.6b under the form of a circle: L (10.13) ρ1 = , 2 L ρ2 = − . (10.14) 2 The corresponding pressure diﬀerences6 are therefore given by 4γ , (10.15) ∆pmax = L 4γ (10.16) ∆pmin = − . L The search for the right ∆p is done within this interval by dichotomic search. 3. Integrate (10.10) with the initial conditions (10.11) and the value of ∆p chosen in point 2. This integration has to be led until the computed meniscus crosses the proﬁle of the gripper, i.e., while z + r tan α − R cos α − R sin α tan α ≤ 0.

(10.17)

This “until” condition is achieved by the events detection embedded in the ode suite of Matlab. 4. Compute the angle made by the meniscus and the gripper (see Fig. 10.7): π − φ + α, (10.18) 2 dr where tan φ = dz . The pressure diﬀerence is adapted by dichotomic end search, i.e., θ2 =

∆pmax ∆pmin

(i) (i)

∆p(i+1) 6

= ∆p(i) ,

if θ2 is to small

= ∆p(i) , ∆pmin =

if θ2 is to large (i)

+ ∆pmax 2

(i)

.

These values have been actually doubled in the simulation code.

(10.19)

10.4 Numerical Model Based on the Laplace Equation

91

z

φ α

θ2 α

B z0

A R

β r r0 Symmetry axis

Fig. 10.7. Determination of the computed value of θ2 z

z d/2

C

V4 V = V1+V2-V3-V 4

D

α

V2

α

V3

B

V1

A

E R

R

β

β

r Symmetry axis

r Symmetry axis

(a)

(b)

Fig. 10.8. Determination of the volume of liquid V = V1 + V2 − V3 − V4 . If there is no internal channel, V4 = 0 (the diameter d vanishes)

While the value if θ2 does not fall within an error interval [θ2 (1 − error), θ2 (1 + error)], reiterate the previous steps from point 3. 5. Compute the volume of liquid corresponding to the value of β chosen in point 1 (see Fig. 10.8): V = V1 + V 2 − V 3 − V 4

(10.20)

92

10 Example 1: Application to the Modeling of a Microgripper z

z

q1

p1

90−b

q1

θ FT

α

C

B FL

p0

α

FT(attractive)

p2

B

A A

FL(repulsive)

β

θ

90−b

FL(attractive)

R

β

R

r r0 Symmetry axis

r0

r

Symmetry axis

(a)

(b)

Fig. 10.9. (a) Closed gripper (no dispensing channel): The tension force is always attractive while the Laplace force (as depicted) is attractive if the pressure inside the liquid is smaller than the surrounding pressure. (b) Gripper with a dispensing channel: If the tightness in C is considered, the Laplace force is repulsive between the symmetry axis and C and becomes attractive between C and A, at least if the curvature of the meniscus is negative (as shown). The tension force is always attractive

with V1 =

n−1

πY (1, i)2 (zi+1 − zi ),

(10.21)

i=1

π Y (1, end)3 tan α, 3 1 2πR3 3 V3 = 1 − cos β + cos3 β , 3 2 2 3 d V4 = π tan α. 24 6. Compute the capillary force. In the case of Fig. 10.9a, the force is given by V2 =

FL = − π(R cos β)2 ∆p, π , FT = 2πR cos βγ cos β + θ1 − 2 F = FL + FT .

(10.22) (10.23) (10.24)

(10.25) (10.26) (10.27)

The sign ‘−’ indicates that a negative pressure diﬀerence ∆p leads to an attractive force. In the case of a gripper fed with a pressure p1 (case of an internal channel), the Laplace term should be replaced by FL = −π(R sin α)2 (p1 − p0 ) − πR2 (sin2 β − sin2 α)∆p,

(10.28)

10.5 Benchmark

93

where in this case ∆p = p2 −p0 . Let us note that an attractive force is counted positive in these equations.

10.5 Benchmark The ﬁrst result will be a validation by benchmarking the numerical model in the case of a cylindrical gripper (α = 0) for which the capillary force is expressed by the Israelachvili approximation [89]: F = 2πRγ(cos θ1 + cos θ2 ).

(10.29)

The force approximation is F ≈ 0.196 mN (see Fig. 10.10). The inﬂuence of the gripper angle α is plotted in Fig. 10.11, from which it can be seen that α inﬂuences the position of the force peak (a ﬁner analysis will be led in next section concerning the signiﬁcance of the apparently inﬁnite peak). The inﬂuence of the coating of the gripper (i.e., the inﬂuence of the contact angle θ2 is shown in Fig. 10.12, from which it can be seen that the force is still attractive with nonwetting coated grippers (i.e., θ2 > 90◦ ), which is interesting for our application. To conclude, it should also be observed that (1) it is almost impossible to work with ﬁlling angles smaller than the gripper angle because the volume to dispense is much too small and the sensitivity of the ﬁlling angle (and therefore of the force) is much too large with respect to a small variation of the dispensed volume of liquid; (2) the force is of the order of 0.1 mN (for a gripper diameter of 0.5 mm). 2

x 10−4 ← FIsraelachvili

1.9

Force [N]

1.8 1.7 1.6 1.5 1.4 1.3

0

10

20

30

40

50

60

Filling angle [°]

Fig. 10.10. Benchmarking of the proposed numerical model (solid line) by comparison with the analytical approximation of (10.29) (square). Let us note that this approximation is valid for a vanishing ﬁlling angle. Value of the parameters: α = 0, θ1 = θ2 = 30◦ , R = 250 µm (or D = 500 µm), d = 100 µm, γ = 72 × 10−3 N m−1 , error on θ2 < 1%

10 Example 1: Application to the Modeling of a Microgripper 4

5

6

4 4 3 2

2

0

1 0

10

20 30 40 Filling angle [°] ◦

0.01

3

0.008

2.5 2

0.006

1.5

0.004

1 0.5

0.002

0 −0.5 0

0 60

50

0.012

3.5

6

8

−2

x 10−12

−3

x 10 7

Force [N]

Volume of liquid [m³]

−12

Force [N]

x 10

10

Volume of liquid [m³]

94

10

20 30 40 Filling angle [°]

0 60

50

(b) α = 10◦

(a) α = 5

Fig. 10.11. Comparison between the numerical (solid lines) and the analytical (dashed lines) models for diﬀerent geometries. The increasing curves with circle marks represent the volume of liquid (left-hand side y label) and the curves with the peak represent the force (right-hand side y label). Value of the parameters: θ1 = θ2 = 30◦ , R = 250 µm (or D = 500 µm), d = 100 µm, γ = 72 × 10−3 N m−1 , error on θ2 < 1% 0.012

0.008

2.5 2

0.006

1.5

0.004

1 0.5

0.002

0 −0.5 0

10

20 30 40 Filling angle [°]

50

0 60

Volume of liquid [m³]

0.01

3

Force [N]

Volume of liquid [m³]

3.5

3

x 10−3 1.4

x 10−12

1.2

2.5

1

2 1.5

0.8

1

0.6

0.5

0.4

0

0.2

−0.5

0

(a) θ2 = 30◦

10

20

30

40

50

Force [N]

x 10−12 4

0 60

Filling angle [°]

(b) θ2 = 110◦

Fig. 10.12. Comparison between the numerical (solid lines) and the analytical (dashed lines) models for diﬀerent coatings. The increasing curves with circle marks represent the volume of liquid (left-hand side y label) and the curves with the peak represent the force (right-hand side y label). Value of the parameters: α = 20, θ1 = 30◦ , R = 250 µm (or D = 500 µm), d = 100 µm, γ = 72 × 10−3 Nm−1 , error on θ2 < 1%

10.6 Pressure Diﬀerence Saturation At ﬁrst sight, the meniscus seems not to be deﬁned for β = α: since the gap between the gripper and the component leads to zero in this point, the pressure diﬀerence given by the Laplace equation (∆p = 2Hγ) leads to inﬁnity as shown in Fig. 10.13. In this ﬁgure, the force (circle marks) is maximum for β = α = 20◦ (by the way, let us note the respective contributions of the tension term (− − − − −) and laplace term (triangular marks, which

10.6 Pressure Diﬀerence Saturation 3

95

x 10−12 0.01 0.009

2.5

0.007

2

0.006 1.5

0.005 0.004

1

Force [N]

Volume of liquid [m³]

0.008

0.003 0.002

0.5

0.001 0

0

10

20

30 Filling angle [°]

40

50

0 60

Fig. 10.13. Force as a function of the ﬁlling angle β (D = 0.5 mm, α = 20◦ , θ1 = 8◦ , θ2 = 50◦ , γ = 72 mN m−1 ). Solid line: volume of liquid, ———–: tension term, triangles: laplace term, circles: total capillary force, · · · · · · : limiting value due to cavitation (Reprinted with permission from [112]. Copyright 2006 Institute of Physics.) 3

x 10−12

x 10−3

Shift

3.5 3

2

2.5 2

1.5

1.5

Force [N]

Volume of liquid [m³]

2.5

1 1 0.5 0

0.5

0

10

20

30 Filling angle [°]

40

50

0 60

Fig. 10.14. Shift of the maximal force due to the pressure diﬀerence saturation (D = 0.5 mm, α = 20◦ , θ1 = 8◦ , θ2 = 50◦ , γ = 72 mN m−1 ). Solid line: volume of liquid, − − −−: tension term, dots: Laplace term, circles: total capillary force (Reprinted with permission from [112]. Copyright 2006 Institute of Physics.)

96

10 Example 1: Application to the Modeling of a Microgripper

are superposed here with the circle marks)). Nevertheless, the pressure difference cannot be unlimited, since the external pressure is assumed to be the atmospheric pressure. Consequently, the pressure diﬀerence used in the force calculation cannot be larger than 101,400 Pa, or even less if the cavitation of the liquid is assumed to occur at the vapor pressure. For example, in the case of water at 20◦ C, the vapor pressure is equal to 2,839 Pa. For isopropanol, the vapor pressure at 20◦ C is equal to 44,029 Pa. Therefore, the Laplace force cannot exceed a value FM = Area× (ambient pressure−vapor pressure). This limiting value is plotted in Fig. 10.13 (· · · · · · ). This leads to a shift of the maximal force as illustrated in Fig. 10.14.

10.7 Conclusions This chapter has illustrated on a watch bearing related case study the arc circle approximation and the numerical approach in the preceding chapters. As a summary of this chapter, let us note that the developed force is about 100 µN for a 500 µm diameter gripper, that this force is increasing with decreasing volumes of liquid (however behind the typical force peak), that the force still remains attractive in the case of moderate non wetting grippers (θ2 = 110◦ ). The experimental characterization of the involved parameters will be described in Chap. 19 and gripping experiments will be lead in Chap. 20.

11 Second Set of Parameters

11.1 Introduction This chapter presents more advanced aspects related to the capillary force modeling. In particular, as we have seen in Fig. 10.3, the contact angle which is used as an input of the model depend on several parameters such as for example surface heterogeneities, surface impurities, and surface roughness. Moreover, equilibrium state has been assumed in all the force models. The limitations implied by this assumption will be considered in Chap. 12, but the damping eﬀect occurring when considering the dynamics of the problem will be brieﬂy considered in this chapter.

11.2 Surface Heterogeneities and Surface Impurities Let us assume a heterogeneous surface containing two materials 1 and 2. A fraction f1 of this surface is characterized by a surface energy leading to a contact angle θ1 and the other part of the surface (fraction f2 = 1 − f1 ) leads to the contact angle θ2 (Fig. 11.1a). The theoretical contact angle given by the Young equation (6.1) is modiﬁed into an eﬀective contact angle θC given by the Cassie equation [1, 90]: cos θC = f1 cos θ1 + f2 cos θ2 .

(11.1)

Another expression has been proposed by [90] but it seems that for the same values of θ1 , θ2 , f1 , and f2 , it will always predict a smaller contact angle than that obtained with (11.1) (1 + cos θC )2 = f1 (1 + cos θ1)2 + f2 (1 + cos θ2 )2 .

(11.2)

98

11 Second Set of Parameters qR

θΑ Hysteresis

Hydrophilic region leads to a receding angle smaller than expected

Hydrophobic region leads to an advancing angle higher than expected

(a)

fi qi

Model

qC

(b)

Fig. 11.1. Inﬂuence of surface impurities or surface heterogeneities: (a) Macroscopic example of a smooth heterogeneous surface [133], illustrating the advancing contact angle θA when the meniscus is about to move downward and the receding contact angle θR when the meniscus is about to move upward. The contact angle hysteresis is equal to θA − θR ; (b) Schematic model to modify contact angles

11.3 Surface Roughness Let us assume a droplet placed on a rough substrate: Due to the roughness asperities, the actual area is bigger than the apparent one. Let us now introduce δ the ratio of the actual interface area to the apparent one. The area of the actual (i.e., rough) area of the solid–vapor (solid–liquid) interface is denoted by ASV (ASL ). The apparent surface is a projection of the rough surface (see Fig. 11.2): δ=

ASL ASV = . AApparent AApparent

(11.3)

Using δ, (6.2) can now be rewritten into ∆G = δ ∆AApparent γSL − δ ∆AApparent γSV + ∆AApparent γ cos θ . (11.4) lim∆A→0 ∆G ∆A = 0 Combining (11.4) and the expression of the contact angle given by the Young equation, the eﬀective contact angle θrough can be expressed as a function of the surface ratio δ and the contact angle θsmooth made of the liquid on a plane smooth substrate made of the same material: cos θrough = δ cos θsmooth .

(11.5)

This approach was ﬁrst proposed by Wenzel and more detailed information can be found in [1] and [73]. Henceforth, (11.5) can feed the previous simulation with contact angles corresponding to actual rough surfaces. That is important if the simulation is used to design gripper tips that usually present roughness proﬁles. From (11.5), we see that angles lower than 90◦ are decreased by roughness, while the angle increases if θ is larger than 90◦ . This means that the gripping force of a capillary gripper could be increased by using rough gripper tips.

11.4 Static Contact Angle Hysteresis γ

99

Rough surface

vapour

liquid γSL θ

Apparent surface

gSV Projection lines

solid

(a)

(b)

δ qsmooth

Model

qrough

(c)

Fig. 11.2. Inﬂuence of surface roughness: (a) Contact line on a rough substrate; (b) Actual and apparent surfaces; (c) Model to modify contact angles

It must be noted that surface roughness can lead to condensing humid air in small cavities of the surface and hence to an attractive force Lcp due to liquid bridging [103]: Al γ , (11.6) Lcp = rk where Al is the surface area where meniscus formation occurs and rk is the Kelvin radius given by the Kelvin equation [1]1 : γv rk = , (11.7) RT log (p0 /p) where v is the molar volume of the liquid, R is the perfect gas constant, T is the absolute temperature, and p0 /p is the relative vapor pressure (= relative humidity for water). Israelachvili [89] gives γv/RT =0.54 nm for water at 20◦ .

11.4 Static Contact Angle Hysteresis When the contact line is about to move, one observes contact angle changing. The receding angle is smaller than the static angle while the observed angle, when moving forward, is larger than the static contact angle. A model has been proposed by Zisman (see Adamson [1]) who observed that cos θA (advancing angle) is usually a monotonic function of γ. Henceforth, he proposed the following equation: cos θA = a − bγ

(11.8)

Reference [73] cited [95] for a detailed study of the eﬀect of roughness on contact angle hysteresis. This hysteresis implies that even at equilibrium, the contact angle value is not unique. A way to improve the static simulation can be to use the receding contact angle during the picking task: indeed, at least at the beginning of the picking task, the object is in contact with the gripper and the meniscus spreads out over its maximum area. Consequently, during the vertical translation of the gripper, the meniscus is supposed to retract (see Fig. 11.3). Hence, the contact angle value should be that of the receding angle. Nevertheless, experiments show that the actual angle is closer to the advancing contact angle: This will be discussed in Sect. 18.3. 1

log = loge = ln = log10 .

100

11 Second Set of Parameters

(a) Advancing contact angle θA

(b) Receding contact angle θR

Fig. 11.3. Contact angle hysteresis 3 2.5 log θ

(180°) 2 1.5 1 0.5 −5

−4

−3

−2 −1 log Ca

0

1

2

Fig. 11.4. Dynamic contact angle θ as a function of the capillary number Ca = µV (µ=dynamic viscosity (Pa s), V =velocity of the contact line (m s−1 )) for several γ silicone oils ﬂowing in a 1.955 mm diameter glass capillary. (Reprinted from [84], with permission from Elsevier)

11.5 Dynamic Spreading Unfortunately, the contact angle hysteresis cannot be put aside only by choosing the advancing or the receding angle, depending on the relative motion of the object and the gripper. Indeed, the contact angle also depends on the velocity of the contact line. This phenomenon is described in [84]. We illustrate it with their results in Fig. 11.4. According to a representation in [46], the results are presented in the logarithmic scale, showing the law: √ 3 (11.9) θ ≈ V, where θ is the dynamic contact angle and V is the velocity of the contact line. More information on the way to model the contact angle will be searched in order to use it in the simulation. We see that the problem will be more complex than just feeding the static simulation with diﬀerent contact angles as those suggested in sect. 11.4: The spreading of the liquid bridge is a dynamic process. Reference [71] presents results of a force measure during the approach of a tip with a 0.28 µL droplet hung on it: We can see the evolution of this force in Fig. 11.5. An interesting observation is that during phase C (A, B, and C are included in the approach phase of the tip toward the surface),

11.6 Conclusions

101

Force (mN)

20

10

0 6

8

10

12

14

16

18

20

22

Time (s)

Fig. 11.5. Force as a function of time: repulsive force brieﬂy occurs during phase C (A, B, C = approach phase) – Water droplet (volume=0.28 µL on a Si chip [72]): This eﬀect highly depends on the volume of liquid and is not observed with a 0.19 µL volume. (Reprinted with permission from [72], Copyright 2002 Springer)

the capillary force becomes repulsive: Beneﬁt can be taken from this positive pressure in a capillary gripper because this damping eﬀect prevents damage at contact, which is one of the main drawbacks of a vacuum or mechanical gripper.

11.6 Conclusions If roughness and impurities can be taken into account by correcting the contact angle with (11.1), (11.2), and (11.5), dynamic eﬀects cannot be obtained by a static simulation: It is one of the perspectives to develop a dynamic simulation. Moreover, a companion topic linked to these simulations is the problematics of capillary condensation. This problematics can be divided into two problems: –

–

The capillary condensation at the tip of a surface force apparatus (SFA): A modeling of the condensation menisci based on the arc approximation is presented in [161]. Our static simulation could be adapted to this problem easily because only one iteration loop is required on the starting position of the liquid bridge in order to adjust the contact angle θ1 . Indeed, ∆p is directly given by (11.7). The capillary condensation due to roughness of the microsystems. This problem is addressed in [176] and in [126]. These authors investigated this aspect because of its implications on the reliability of microsystems but we have not found any study of this subject in relation with micromanipulation or microassembly.

The problematics of capillary condensation has been pointed out as a source of MEMS breakdown [104, 125, 179, 184] and has been applied to microassembly by [36, 33].

12 Limits of the Static Simulation

12.1 Introduction This chapter aims at validating the static simulation at constant volume in the limits of an assembly case study. Basically, the static simulation cannot aﬀord to give information on the typical cycle time of an surface tension based assembly task (in other words, since the time is not a parameter of the static simulations, the latter cannot output any characteristic time). Therefore, some approximations will be tempted using dimensional analysis and a 1D analytical model derived from the so-called Lucas–Washburn equation. Moreover, high accelerations are usually applied in assembly machines (up to 10g): therefore, it should be checked whether this acceleration can deform the equilibrium meniscus shape or not.

12.2 Performances of the Assembly Machines We will denote by a the typical acceleration of an assembly machine, v its typical velocity, τpick and τplace the typical pick and place operation times. Values are given in Table 12.1 (source: [174]). As far as acceleration is concerned modern linear motors [61] reach up to 100 m s−2 .

12.3 Nondimensional Numbers and Buckingham π Theorem An adimensional analysis has been achieved, involving all the parameters deﬁned in Table 12.2. Knowing the SI dimensions of each dimensional parameter, the dimensional matrix D can be built, whose element Dij is the exponent of the jth dimension in the ith parameter:

104

12 Limits of the Static Simulation

Table 12.1. Typical values of an assembly machine a 20 m s−2

v 1.5 m s−1

τpick 60 ms

τplace 60 ms

Table 12.2. List of the parameters involved in the simulation (Ci =coeﬃcient) Ci Parameter M L T C1 µ 1 −1 −1 γ 1 0 −2 C2 C3 d 0 1 0 γSL 1 0 −2 C4 γSV 1 0 −2 C5 F 1 1 −2 C6 a 0 1 −2 C7 C8 v 0 1 −1 r 0 1 0 C9 ρ 1 −3 0 C10 C11 g 0 1 −2

Description Dynamic viscosity of the used liquid Surface tension Characteristic size of the meniscus Interfacial energy solid–liquid Interfacial energy solid–vapor Force exerted by the meniscus Gripper acceleration Gripper characteristic velocity Gripper travelling range Density Gravity

⎛

1 −1 ⎜1 0 ⎜ ⎜0 1 ⎜ ⎜1 0 ⎜ ⎜1 0 ⎜ D=⎜ ⎜1 1 ⎜0 1 ⎜ ⎜0 1 ⎜ ⎜0 1 ⎜ ⎝1 −3 0 1

Value 1–50 mPa s 20–72 mN m−1 10−4 m

100 m s−2 0.1–0.3 m 1000 kg m−3 9.81 m s−2

⎞ −1 −2⎟ ⎟ 0⎟ ⎟ −2⎟ ⎟ −2⎟ ⎟ −2⎟ ⎟. −2⎟ ⎟ −1⎟ ⎟ 0⎟ ⎟ 0⎠ −2

These n = 11 dimensional parameters can be grouped to form m = n − k adimensional numbers, where k states for the rank of the matrix D [162]. These adimensional numbers πi , (i = 1 : 10) are written as follows: C4 C5 C6 C7 C8 C9 C10 C11 πi = µC1 γ C2 dC3 γSL γSV F a v r ρ g ,

(12.1)

where the n = 11 coeﬃcients Ci are the elements of a 11 × 1 vector C = (C1 , C2 , ..., C11 )T , given by the solutions of the following equation [162]: DT C = 0.

(12.2)

3×11 11×1

It means that each vector C is a vector of the kernel of the matrix D, which can be for example obtained using the Matlab command null(D,’r’). In this case, this leads to 8 = 11 − 3 adimensional numbers (the rank of the matrix D is equal to 3):

12.3 Nondimensional Numbers and Buckingham π Theorem

π1 = π2 = π3 = π4 = π5 = π6 = π7 = π8 =

105

γSL , γ γSV , γ F , γd µ2 da , γ2 µv , γ r , d γdρ , µ2 µ2 dg . γ2

These eight π terms can be recombined with one another, leading to eight other formulations, among which: (π5 .π7 ) → Re = (π52 .π7 ) → We = (π52 /π7 ) → Fr = (π4 .π7 ) → Be = (π7 .π8 ) → Bo = = (π2 − π1 ) → cos θ = τ = =

Inertial eﬀects ρdv = , µ Viscous eﬀects Inertial eﬀects v , = Surface tension eﬀects γ/ρd v Intertial eﬀects √ = , Gravity eﬀects dg Inertial eﬀects aρd2 = , γ Surface eﬀects Gravity eﬀects dρg = γ/d Surface tension eﬀects Hydrostatic pressure , Laplace pressure γSV − γSL = Young–Dupr´e equation γ µd/γ µv = d/v γ Characteristic time of the droplet . Characteristic time of the manipulation

(12.3) (12.4) (12.5) (12.6)

(12.7) (12.8)

(12.9)

These results represent the classical adimensional numbers (Reynolds, Weber, Froude and Bond number). Be is called “Bonding Eﬀect Number” [19]. The term µd/γ represents the characteristic droplet time, which is the time taken by the droplet to move its interface along the characteristic distance d. By comparison, the term vd represents a characteristic manipulation time at

106

12 Limits of the Static Simulation

release: indeed, if the object is being released, the meniscus is stretched over a characteristic length d at the gripper velocity v. However, if the object is not being released (that means that object and gripper remain in contact), the manipulation time must be considered the one given in Table 12.2. Finally, 2 the term ρdµ has the dimensions of a time and gives an order of magnitude of the damping time of the transitory eﬀects occurring inside the droplet.

12.4 Another Approach: Use of a 1D Analytical Model A ﬁrst approximated description can be performed through a simpliﬁcated 1D geometry shown in Fig. 12.1. The capillary rise h of a liquid between two inﬁnite parallel plates separated by a distance d can be described by the following equation: ˙ 12µhh˙ d(hhdρ) =− + 2γ cos θ − hdρg + po d, dt d

(12.10)

where h and h˙ are the position and the velocity of the liquid–gas interface, d is the separation distance between the parallel plates, ρ is the liquid density, µ its dynamic viscosity, γ its surface tension, g the gravity acceleration, and po the pressure diﬀerence between points A and B. The left-hand side of this equation states for the time derivative of the liquid momentum, i.e., the time derivative of the mass×velocity product, where h˙ is the interface velocity and hdρ is the liquid mass involved in the motion. This description is a bit similar to the case in which a chain, initially at rest on a substrate, is pulled upward: the mass of the system is increasing with time, and the pulling velocity can vary. The ﬁrst term on the right-hand side represents the viscous force, working against the motion: it can be estimated from a Hagen–Poiseuille stationary ﬂow. The velocity distribution is parabolic:

θ θ

B L

h d

d A

(a) Fig. 12.1. 1D analytical model

(b)

h

12.4 Another Approach: Use of a 1D Analytical Model

v=−

1 dp y(d − y), 2µ dx

107

(12.11)

where dp/dx is the (constant) pressure gradient along the motion axis, d is the separation distance between both parallel plates, and y is the coordinate perpendicular to the motion direction x (therefore 0 ≤ y ≤ d). Integrating this velocity proﬁle on a section perpendiular to the ﬂow we can get the mass ﬂow: d ρv dy = · · · = −

Q=

ρd3 dp . 12µ dx

(12.12)

0

Hence we can express the pressure gradient as a function of the mean velocity v¯ = Q/(ρd): −

dp 12µ¯ v = 2 . dx d

(12.13)

Since the mean velocity is equal to h, the viscous force can be determined as follows: F =−

12µhh˙ dp hd = . dx d

(12.14)

The second term on the right-hand side is the driving surface tension term, equal to the surface tension projected along the motion direction and acting along the perimeter of the triple line. In this case, this term (as the all equation) is expressed as a force per unit length. The third term is the weight of the volume of liquid and the last term represents the force exerted by the pressure diﬀerence between A and B: It can be considered as a driving or a resistance term depending on its sign. Now that (12.10) is explained, we can divide it by ρd: ˙ 12µhh˙ po 2γ cos θ d(hh) =− − gh + . + 2 dt ρd ρd ρ

(12.15)

In [156], Shoenfeld proposes the following variables substitution: z = h2 → z˙ = 2hh˙

(12.16)

leading to replace (12.15) by: √ 1 6µz˙ 2 cos θ po z¨ = − 2 + + − g z. 2 ρd ρd ρ

(12.17)

The classical analytical approach consists in determining the equilibrium position from (12.17) and in studying small oscillations around this equilibrium position. The equilibrium position is given by:

108

12 Limits of the Static Simulation

ho =

po 2γ cos θ + . ρgd ρg

(12.18)

Around this equilibrium position, we consider small oscillation of z: z = zo + δz,

(12.19)

where zo = h2o . Equation (12.17) becomes: δ¨ z+

√ gδz 4γ cos θ 2po 12µδ z˙ + − 2g zo , +√ = 2 ρd zo ρd ρ

(12.20)

=0

which is similar to: x ¨ + 2λx˙ + ω02 x = 0,

(12.21)

whose solutions are: λx0 + v0 sinh(ωt)) if λ2 > ω02 , (12.22) ω λx0 + v0 sin(ωt)) if λ2 < ω02 , (12.23) x(t) = exp(−λt)(x0 cos(ωt) + ω where ω = |λ2 − ω02 | and, in both cases, x(t = 0) = x0 and x(t ˙ = 0) = v0 . √ In our case, λ = 6µ/ρd, ω02 = g/ z0 , and the initial conditions are: x(t) = exp(−λt)(x0 cosh(ωt) +

δz(t = 0) = δzo , δ z(t ˙ = 0) = 0.

(12.24) (12.25)

We conclude that the characteristic damping time is equal to (see also Fig. 12.2) τ=

1 ρd2 = . λ 6µ

(12.26)

We see that the expression ρd2 /µ and (12.26) diﬀer from a factor 6.

12.5 Limitations of the Static Model In order to set limitations to the proposed constant volume static simulation, characteristic times have been compared and the constant velocity or accelerated motions have been analyzed. –

For water (ρ = 1,000 kg m−3 , γ = 72 N m−1 , µ = 1 mPa s and d = 100 µm), the characteristic times have been determined (Table 12.3.)

12.5 Limitations of the Static Model

109

x 10−4

4

dh [m]

2 0 Damping time

−2 −4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time [s] x

10−3

h [m]

6 4 2 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time [s]

Fig. 12.2. µ = 0.001 Pa s, ρ = 1,000 kg m−3 , γ = 0.072 Nm−1 , θ = 60◦ , d = 0.001m, τ = 0.16667 s Table 12.3. Comparison between the diﬀerent characteristic times (ms): The damping time given by (12.26) is six times smaller, i.e., 1.6 ms, but this is only an order of magnitude Time (ms) Water µd/γ 0.001 ρd2 /µ 10 τpick or τplace 60

–

–

If the whole system including the component, the meniscus, and the gripper are moving at constant velocity, the static description at constant volume is of course valid. However, it can be assumed that there are residual vortex, leading to velocities inside the liquid, and hence to pressure gradient that could modify the capillary force computed by the static simulation. These transitory eﬀects are damped after a characteristic time ρd2 /µ(< τpick , τplace ). In the case of accelerated motions, inertial eﬀects must been taken into account: the object will stick to the gripper only if the acceleration of the gripper a is lower than the limit acceleration a ˜ given by: a ˜=

F − g, m + mliq

(12.27)

110

12 Limits of the Static Simulation

where F is the capillary force, m the mass of the component, and mliq the mass of the liquid. The whole system is now in a motion with constant acceleration and the situation could be described at rest in a moving coordinates system with constant acceleration. In this case, the interface shape could be aﬀected by this acceleration. In the same way as gravity eﬀects on the interface shape are neglected if the characteristic length of the meniscus is smaller than the capillary length Lc = γ/ρg, the eﬀects of the constant acceleration of the gripper are negligible if the characteristic than the modiﬁed capillary length size of the meniscus is smaller γ γ −2 Lc = (a+g)ρ . For a = 10 ms , Lc = (a+g)ρ ≈ 0.8 mm.

12.6 Conclusions As a conclusion of this chapter, it can be stated that the characteristic time of a surface tension based picking task could have typical cycle times down to a few milliseconds. The second conclusion is that equilibrium is a reasonable assumption, even in high throughput machines (a = 10g), as long as the characteristic size of the meniscus is smaller than about 0.8 mm.

13 Approaching Contact Distance, Rupture Criteria, and Volume Repartition After Separation

13.1 Introduction According to the volume of liquid and the shape of the gripper, the hanging droplet will be transformed into a liquid bridge as soon as the separation between the gripper and the component is smaller than zA , the approaching contact distance. The ﬁrst problem treated in this chapter is to compute this distance according to the gripper geometry and the volume of liquid V . The two other problems considered in this chapter are on the one hand the determination of the separation distance (also called the gap) at which the meniscus will break, and on the other hand, the residual volume of liquid remaining on the component after this rupture.

13.2 Approaching Contact Distance The general situation is depicted in Fig. 13.1: a droplet hangs below the gripper, the bottom of it being at a distance zA from the apex of the gripper. If gravity is neglected, this droplet has the shape of a spherical cap deﬁned by its radius R and the angle α (note that the volume of the gripper Vgripper located below y must be subtracted from the cap volume, given in Appendix B.1). First, the intersection point P between the meniscus and the gripper must be found: P is deﬁned by its coordinates (x, y) (the origin of this coordinates frame is located at the gripper apex, O). To do so, the position of P is iterated until the meniscus volume calculated corresponds to the prescribed volume V : this is illustrated in Fig. 13.2. Once x is chosen, y can be determined as a function of the gripper equation. Then, the mathematical relations used in Fig. 13.2 are given below (the expression depending on α in (13.5) is detailed in Appendix B.1): tan φ|(x,y) =

dy |(x,y) dx

(13.1)

112

13 Approaching and Rupture Distances

φ θ α

R

α

Vgripper zA

V

P

φ y

O

x

Fig. 13.1. Approaching contact distance—general case x y = y (x,gripper)

f = f (x,y) α = α (x,y,q)

Vgripper =V (x,y,gripper)

R = R(α,x) V = V(R,α) - Vgripper No V=Vprescribed

zA

Fig. 13.2. Approaching distance algorithm

α(x, y) = θ + φ(x, y) x R= sin α Vgripper = V (φ, gripper) 1 3 2π 3 R 1 − cos α + cos3 α − Vgripper V = 3 2 2

(13.2) (13.3) (13.4) (13.5)

where φ states for the gripper slope at P and θ is the contact angle . At the end of each iteration, the computed volume corresponding to the chosen point P

13.3 Rupture Distance and Residual Volume of Liquid

113

is compared to the prescribed volume V : if it is smaller (larger), x is increased (decreased). Finally, the approaching contact distance is determined by zA = R(1 − cos α) − y.

(13.6)

Note that in the case of the conical gripper, φ is equal to the gripper parameter p, and consequently, α = θ + p is constant. If moreover p = 0 (cylindrical gripper), Vgripper ≡ 0 and R is directly given by (13.5) so that zA is computed in only one iteration. The approaching contact distance is one of the design elements of a capillary gripper: it gives partial information on the required range of the gripper. Other elements are the receding distance at which the meniscus breaks into two separated droplets.

13.3 Rupture Distance and Residual Volume of Liquid The problem of the rupture criterion and the volume repartition can be expressed as follows: 1. The expression of a rupture separation distance z ∗ at which the liquid bridge will be replaced by two droplets adhering to the gripper on the one hand and to the component on the other hand 2. The repartition of the meniscus volume V into the two residual volumes V1 and V2 of the droplets after rupture (see Fig. 13.3) These questions must be addressed both to design the gripper kinematics (what range should the gripper move in order to break the liquid bridge?) and to investigate the suitability of the gripping principle (what volume of liquid still remains on the component after rupture? The answer to this question can z

z

V2

V2

V1

V1 r

(a)

r

(b)

Fig. 13.3. The volume repartition. (a) Meniscus before break; (b) corresponding droplets after break

114

13 Approaching and Rupture Distances

possibly lead to the computation of the evaporation time if the used liquid is volatile enough – which is the case with water, but not with silicone oil). In this chapter, the repartition of liquid will be studied from both an energetic and a geometrical point of view: the energetic approach consists in determining the repartition that extremes the interfacial energy of the system. Unfortunately, it will be shown that this extremum is actually a maximum and moreover the volume repartition does not agree with the experimental results in the case of nonsymmetric conﬁguration (not equal contact angles). Therefore, a second approach called geometrical approach – already proposed in [142] – will be followed and implemented. Once the repartition after breaking is known, the breaking gap can be evaluated by energetic considerations: it will be assumed that the bridge turns itself toward two droplets when the total interfacial energy of the liquid bridge equals that of the two droplets. According to Pepin et al. [142] the energy will be evaluated only at the liquid–vapor interface. As it will be shown that this criterion is inexact, we will propose to correct it by taking into account the interfacial energy of all interfaces: liquid–vapor, solid–vapor, and solid–liquid.

13.4 Mathematical and Notation Preliminaries 1. The volume V and surface S of the portion of a sphere characterized by a radius R and an angle θ are given by the following equations (see Appendix B.1): 3 2πR3 a) Vi = 3 i (1 − 32 cos θi + cos2 θi ) b) Si = 2πRi2 (1 − cos θi ) 2. The volume V is constant: • for analytical developments, the meniscus is assumed to be of cylindrical shape V = πr2 z • the volume of the two spherical caps after rupture is given by V = V1 + V2 ⇒ dV2 = −dV1 3 1 3 3 3. Ai ≡ 2π 3 (1 − 2 cos θi + 2 cos θi ) (As Ai R represents the volume of a spherical cap with radius R and limited by an angle θi , Ai ≥ 0) 4. αi ≡ 16 (2 − cos θi − cos2 θi ) ≥ 0 i 5. A αi = 2π(1 − cos θi ) 2 α1 3 6. K ≡ A A1 ( α2 ) 2 7. K = A A1 8. 2 cos θ ≡ cos θ1R + cos θ2R 9. γSVi = γSLi + γ cos θi 10. θi = contact angle made by the ith droplet, usually the advancing contact angle

13.5 Volume Repartition

115

13.5 Volume Repartition Two models will be studied: 1. Energetic model For small gaps the interfacial energy of the meniscus W is smaller than the interfacial energy U of the two droplets conﬁguration (basically because the area of the meniscus is smaller than that of two droplets with equivalent volume). By increasing the gap, W can become larger than U , leading to the rupture of the meniscus. Once the meniscus is broken, the liquid volume is split into two spherical caps adhering to the component and to the gripper. The underlying idea of this method consists in searching for a repartition of the volumes V1 and V2 that (if it exists) extremes the interfacial energy U (let us note U ∗ this extremum). Let us note that the interfacial energy is the sum of the energy of the liquid–vapor (LV), liquid–solid (LS), and solid–vapour (SV) interfaces. However, some authors [142] claim to get valid results by considering only the energy of the LV interface. Consequently, both assumptions will be considered and the notations related to the second one will be added with the “LV” subscript. The following results are shown in Appendix A.2 for a cylindrical gripper: a) In the case of the LV interface, V (13.7) V1 = 1 + K KV V2 = (13.8) 1 + K (13.9) θ 1 > θ 2 ⇒ V 1 < V2 2/3 2 2 A1 α2 V A2 α1 ∗ = γ + . (13.10) ULV α1 α2 α23 A1 + α13 A2 (13.11) b) In the case of the whole interface, V (13.12) V1 = 1+K KV V2 = (13.13) 1+K θ 1 > θ 2 ⇒ V 1 > V2 (13.14) ∗ 2/3 1/3 (13.15) U = 3γV (A1 + A2 ) . We see from Figs. 13.4 and 13.5 that V1 /V = 0.5 for symmetric conﬁgurations (i.e., θ1 = θ2 ). However, we can conclude from (13.14) that the proposed method (at least in the case of the total interfacial energy model) cannot predict the volume repartition in a suitable way because experiments show that the larger the contact angle the smaller the corresponding volume. Maybe the trend derived from the LV model could ﬁt the experimental

116

13 Approaching and Rupture Distances V1/V 5

0.

99

82

0.64

0.5 49 91

973

0.699 64

6

0.749 55

0.7994

70

0.84937

0.94919

0.89928

80

0.5

60 928 493 7 0.7 994 6 0.7 49 55 0.6 99 0. 64 59 98 0. 64 2 97 3

30

1 99

54

3

0.

10

0.8 0.98928 0.4793 7 09.9746 4 0.955 69 0.964 64 97

20

99

54

0.

09 50 0.4 8 001 0.4

0.8

0.89

40

5

0.

1

0.949 19

θ2 [°]

50

009

0.45

7 502 0.3 036 0 .3 0

5 0. 9 00 8 5 0.44001 0.

5 27 50 504 2 0.30036 0.2 0054 98 0.2 5063 .3 0 9 5 0.1 0. 0.5 2 0.1007 098 50001 00..44500455643 2.10500072 0.050808 00.0.2 0.1

10

20

30

27

0.350

0.30036

45 0.250 4 0.2005 0.15063 0.10072

0.050808 0.050808

40 θ1 [°]

50

60

70

80

Fig. 13.4. Ratio V1 /V as a function of contact angles θ1 and θ2 after rupture of the meniscus (cylindrical gripper) – Case of the LV interface

3

0.

25

0.

0.

35

V1/V

80

5

4

0.

0.2 5

0.1

0.

45

0.

0.1

0.0

5

70

5 0.6 0.7

25 0. 3

60 0.

2 0.

5 0.0

40

0.

25

30

.55 5 06 0. 50. 0..76 0 5 5 5 0.95 3..43.4 0.87 0.000 5 0. .8 00.9

0.0 0.51 0. 15

10

0. 2

20

10

20

3 35 0. 00..40.45

5 5 .5 0. 00.6 5 6 0..7 0 5 0..78 0

5 0.8 0.9

30

5 0.7 0.8

5 0.55 0.6 5 0.6 0.7 5 0.7 0.8

0.

35 0. 4 5 0. 0.4

0.1 0. 15

θ2 [°]

50

5 0.5 0.6

0.85

0.9

.85

0 0.9

0.95

0.95

40

50

60

70

80

θ1 [°]

Fig. 13.5. Ratio V1 /V as a function of contact angles θ1 and θ2 after rupture of the meniscus (cylindrical gripper) – Case of the total interface

observations, but unfortunately the total energy cannot be reduced to the LV energy. Consequently, the suggested method is not suitable to predict the volume repartition. Nevertheless, let us keep in mind the values of U ∗ ∗ and ULV : it will be later investigated if the comparison of the meniscus ∗ energy W (WLV ) with U ∗ (ULV ) provides a suitable rupture criterion, in a

13.6 Rupture Condition and Rupture Gap

117

z

V2 V1 ∆z

zneck ri

r

Fig. 13.6. Repartition of the volumes above and below the meniscus neck

symmetric conﬁguration with θ1 = θ2 (in this case, the predicted volume repartition corresponds to the trivial repartition V1 = V2 = V2 ). Otherwise, we will conclude that this method is never suitable. 2. Geometrical model The geometrical model proposed in [142] assumes that the repartition of the volume between the two droplets after rupture is equal to the distribution of the liquid above and below the meniscus neck right before rupture occurs (Fig. 13.6): V1 =

i=i neck

πri2 ∆z

(13.16)

i=1

where the index ineck is deﬁned by rineck = min(ri ). This method for volume repartition is used in the Sect. 13.6 to determine V1 and V2 .

13.6 Rupture Condition and Rupture Gap On the one hand, the interfacial energy W (z) of the meniscus increases as a function of the gap; on the other hand, an interfacial energy U (V1 , V2 ) can be associated to the volume repartition of the two droplets after rupture. The idea in [142] consists in equaling both energies to get a rupture criterion allowing to assess the rupture gap z ∗ : W (z ∗ ) = U (V1 , V2 )

(13.17)

Two ways can be followed to evaluate the interfacial energy: 1. The LV interfacial energy approach proposed in [142] takes only the LV interfacial energy WLV into account. WLV can be written as a function of the surface tension γ and the lateral area of the meniscus Σ:

118

13 Approaching and Rupture Distances

WLV = γΣ 2 2π z2 dr =γ dφ r 1 + dz dz z1 0 2 n−1 ri+1 − ri ≈ γ2π∆z ri 1 + . ∆z i=1 The developments of Appendix A.2 lead to 1/3 1/3 A1 A2 2/3 2/3 ULV (V1 , V2 ) = γ V + V α1 1 α2 2

(13.18) (13.19)

(13.20)

(13.21)

The gap z ∗ for which the meniscus can break into two residual droplets is given by the condition WLV (z ∗ ) = ULV (V1 , V 2).

(13.22)

2. The total interfacial energy approach we propose consists in adapting the criterion of [142] by considering the total interfacial energy W . By assuming that the sum of the liquid–vapor area ALV and the solid–vapor area ASV is constant and by taking the Young–Dupr´e equation into consideration, W can be expressed as follows: W = γΣ + γSL1 ASL1 + γSV1 ASV1 + γSL2 ASL2 + γSV2 ASV2 = γΣ + ASL1 γSL1 + (C1 − ASL1 )γSV1 + ASL2 γSL2 +(C2 − ASL2 )γSV2 = γΣ + ASL1 (γSL1 − γSV1 ) + ASL2 (γSV2 ) = γΣ − ASL1 γ cos θ1R − ASL2 γ cos θ2R .

(13.23) (13.24) (13.25) (13.26)

Note that the arbitrary energy level has to be chosen so that C1 = C2 ≡ 0. By considering ﬂat components and axially symmetric grippers, W is rewritten into 2 n−1 ri+1 − ri ri 1 + − πr12 γ cos θ1R W = 2πγ∆z ∆z i=1 +2π

j=j m

rj

1 + rj2 ∆z

(13.27)

j=1

where the index jm is so that rjm = r2 , the radius of the intersection point between the gripper and the meniscus. Note that rj = dr(z) dz |j , where r(z) is the gripper proﬁle equation. Once again, developments of Appendix A.2 lead to 1/3

2/3

U (V1 , V2 ) = γ(3A1 V1

1/3

2/3

+ 3A2 V2

).

(13.28)

13.7 Analytical Benchmarks

119

The gap z ∗ for which the meniscus can break into two residual droplets is given by the condition W (z ∗ ) = U (V1 , V2 ).

(13.29)

13.7 Analytical Benchmarks In (13.20) and (13.27), the energy is numerically evaluated. However, if we assume a cylindrical shaped meniscus (Fig. 13.7), the lateral area Σ can be expressed as Σ = 2πrz √ = 2 πzV .

(13.30)

Consequently, (13.20) and (13.27) can be expressed by √ √ √ WLV ≈ 2 π V γ z

(13.31)

√ √ √ 2V γ cos θR W ≈ 2 π Vγ z− z cos θ

(13.32)

+cos θ

1R 2R . By combining, respectively, (13.21) and (13.31) where cos θR = 2 at the one hand and (13.28) and (13.32) at the other hand, the rupture criteria 13.22 and 13.29 can be written as follows:

√ √ ∗ =γ 2 π V γ zLV

1/3

1/3

A1 A 2/3 2/3 V + 2 V2 α1 1 α2

√ √ √ 2V γ cos θR = 3γV 2/3 (A1 + A2 )1/3 . 2 π V γ z∗ − z∗

(13.33)

(13.34)

z

Approximation: cylindrical shape

r

Fig. 13.7. The meniscus volume is approximated by a cylinder for the determination of the lateral area Σ

120

13 Approaching and Rupture Distances

Let us now assume a symmetric case (θ1 = θ2 and the energetic approach for volume repartition gives the trivial and suitable distribution V1 = V2 = V2 ) in order to further simplify these equations: ∗ = zLV

V 1/3 −2/3 2/3 −2 4 A α π

(13.35)

√ √ √ 2V γ cos θrec 2 π V γ z∗ − = 3γV 2/3 (2A)1/3 . (13.36) z∗ Finally, let us consider the trivial case θ1 = θ2 = π/2. This implies α = 1/3 and A = 2π/3, leading to a unique gap: ∗

z =

∗ zLV

=

V π

1/3

81 4

1/3 .

(13.37)

Let us note that for θi = π/2, W and WLV tend to each other because this contact angle corresponds to a neutral situation: the surface does not repel and is not wetted by the liquid.

13.8 Summary of the Methods Several approaches can be combined: Useful information can be summarized as follows: 1. Case “0” ∗

z =

81V 4π

U ∗ = 3V 2/3 2. Case “1” ∗ ULV

=γ

1/3 (13.38)

4π 3

1/3

1/3 γ.

(13.39)

1/3

A1 A 2/3 2/3 V + 2 V2 α1 1 α2

(13.40)

V1 =

V 1 + K

(13.41)

V2 =

K V 1 + K

(13.42)

∗2 ULV . 4πV γ 2

(13.43)

∗ zLV =

3. Case “2” U ∗ = 3γV 2/3 (A1 + A2 )1/3

(13.44)

13.8 Summary of the Methods

121

z U

U* q2

z V2 V1 = 0

r r r

r*

(a)

(b)

r

Fig. 13.8. The algorithm. (a) Iterative search; (b) sketch of the droplets energy as a function of the drop radius r

V 1+K KV V2 = 1+K √ λ ∗ z | µ z∗ + ∗ = U ∗ z V1 =

(13.45) (13.46) (13.47)

√ √ with λ = −2V γ cos θR and µ = 2 π V γ. 4. Cases “3 and 4” The following algorithm has been used (Fig. 13.9a): a) To determine the volume repartition V1 + V2 = V , search for a radius r¯ so that the droplet hanging below the gripper that intersects the r) = V . This means that the corregripper at r = r¯ has a volume V2 (¯ sponding volume V1 for the droplet put on the component is equal to zero, as illustrated in Fig. 13.8a b) For r ∈ [0, r¯], compute V2 (r) c) Deduce V1 = V − V2 (r) and hence U (V1 , V2 ) d) Search for U ∗ = max(U ) and the corresponding radius r∗ , as illustrated in Fig. 13.8b e) Choose a starting gap z, as small as possible (to be sure that W < U ) f) Compute the corresponding meniscus and the associated energy W (z) g) Compare W (z) > U ∗ • If the comparison is true, the rupture gap z ∗ is equal to z • Otherwise, the gap must be increased and the steps [f–g] must be iterated 5. Cases “5 and 6” The following algorithm has been used (Fig. 13.9b): a) Choose a starting gap z as small as possible (to be sure that W < U ) b) Compute the corresponding meniscus

122

13 Approaching and Rupture Distances Search for r | V2(r) = V and V1=0

Choose a gap = z Determine meniscus (z)

Choose a gap=z Determine meniscus (z) Compute zneck

Compute W(z)

Determine V1(z),V2(z),U(z) and W(z)

W>U*

W>U

No: increasegap

Determine r* corresponding to U*=max(U )

No: increasegap

For r C[0,r], compute V1(r), V2(r) and U(r)

Yes

Yes Rupture gap z* = z

Rupture gap z*=z

(a)

(b)

Fig. 13.9. Algorithms used in the general case. (a) Volume repartition determined with the energetic approach; (b) volume repartition determined with the geometrical approach Table 13.1. Summary of the methods ID Volume repartition

Rupture criterion Geometrical conﬁguration V 2

0

Trivial: V1 = V2 =

1 2 3 4 5 6

Energetic approach Energetic approach Energetic approach Energetic approach Geometrical approach Geometrical approach

LV Total LV Total LV Total

Trivial: cylindrical gripper (p = 0) and θ1 = θ2 = π2 Cylindrical gripper (p = 0) Cylindrical gripper (p = 0) General case General case General case General case

c) Determine the corresponding energy W (z) and the position of the meniscus neck and compute the volumes V1 below zneck and V2 above zneck d) Compute the energy U (z) associated to the liquid repartition V1 and V2 e) Compare W (z) > U ∗ • If the comparison is true, the rupture gap z ∗ is equal to z • Otherwise, the gap must be increased and the steps [b–e] must be iterated

13.9 Comparison between the Methods The diﬀerent methods of Table 13.1 have been compared for cylindrical gripper, both for symmetric (θ1 = θ2 ) and nonsymmetric (θ1 = θ2 ) cases. The accuracy required on the volume in the computation of a meniscus was 1%

13.9 Comparison between the Methods

123

0.6

V2/V (squares)

0.5 0.4 0.3 0.2 0.1 0 20

30

40

50

60

70

80

90

100

110

θ1=θ2 [°]

Fig. 13.10. Normalized volumes V2 /V obtained by the diﬀerent methods, between which it can hardly be distinguished gap3

Rupture gap and rupture neck [µm]

2000 gap1 gap5 1500 gap0 1000

z1 z5

gap4 gap6

z3 gap2 z4 ≈ z6

500 z2 0 20

30

40

50

60

70

80

90

100

110

θ1=θ2 [8]

Fig. 13.11. Gap and zneck

and the gripper was discretized with 1,000 points. The imposed volume was V = 0.28 µL and the surface tension γ = 72 mN m−1 . The results of the symmetric case are shown in Figs. 13.10 and 13.11. Figure 13.10 shows the normalized volume V2 /V = 1/2 for all volume repartition methods (in the symmetric case). The rupture gap z ∗ and the rupture neck height zneck are plotted in Fig. 13.11 (respectively represented by the lines

13 Approaching and Rupture Distances Normalized volume V2/V for θ1 = 60° and V = 0.28µL

124

1

0.8

0.6

0.4

0.2

0 20

30

40

50

60 70 θ1 [°]

80

90

100

110

Fig. 13.12. Normalized volume V2 /V as a function of θ1 computed by the energetic approach (stripped line) and by the geometrical approach (continuous line)

with circles and squares): it can be seen that all methods converge toward the same result when the contact angles tend to π/2. For θ < π/2, the methods based on a LV energy computation (dotted curves above the stripped line corresponding to the trivial case 0) lead to unacceptable gaps that can even reach up to 2 mm, which was never observed in the experiments. Moreover, the trend does not seem to be physically acceptable: the gap cannot increase by decreasing the angle, i.e., by increasing the wetting properties. The method of case 2 (dotted-stripped line) is based on analytical developments assuming that the meniscus is of cylindrical shape. Its results are consequently in very good agreement with the numerical methods of cases 4 and 6 (continuous lines) when θ → π/2 but are no longer correct for smaller contact angles. The distinction between cases 4 and 6 requires a nonsymmetric simulation (see results of Fig. 13.12). A nonsymmetric case has been studied for V = 0.28 µL water, with θ2 = 60◦ . Figure 13.12 indicates that V2 (V1 ) is larger (smaller) than 1/2 when θ1 < θ2 , when the energetic repartition method is used: this is not physically correct because smaller θ corresponds to larger adhesion and henceforth larger volumes.

13.10 Conclusions We conclude this chapter by noting that •

The repartition volume method can be based on the geometrical repartition method already proposed in [142]

13.10 Conclusions

•

125

The LV interfacial energy computation is not correct; henceforth, we propose to base the energy computations on all interfaces (LV, SV, and SV)

A ﬁnal remark is that V1 is also the residual volume left on the component after release. It should be noted from a surface tension micromanipulation point of view that this residual volume of liquid might be a problem: in this case, it is suggested to use a volatile liquid such as water or alcohols, whose evaporation leads to the suppression of this residual volume of liquid. A comprehensive description of the evaporation phenomenon is beyond the scope of this work.

14 Example 2: Numerical Implementation of the Proposed Models

14.1 Introduction This chapter aims at presenting the numerical implementation of the proposed models. Therefore, the problem of computing the meniscus shape will be addressed and coupled with a pseudodynamic simulation, in which the component is moved at each time step according to the capillary force acting on it at this time step. This approach relies on the assumption stated in Chap. 12 that even in high acceleration assembly machines, the meniscus shape was hardly modiﬁed by this acceleration, as long as the meniscus height would remain smaller than the modiﬁed capillary length. Figure 14.1 details all the inputs, related to the materials (liquid, material of the component...) and to the user’s choices (volume of liquid, gripper...) while the outputs (meniscus shape and pressure diﬀerence) are used to compute the capillary force and move the component according to the Newton’s motion law.

14.2 Liquid Bridge Simulation for the Analysis of a Meniscus As explained in Chap. 8, the meniscus shape is obtained by a double iterative scheme (Figs. 14.2 and 14.4). First, a so-called “starting point” is chosen along the gripper proﬁle: this point is supposed to be the ﬁrst point of the meniscus and moreover the slope of the gripper at this point and the contact angle are known, providing initial conditions for the meniscus determination. For this starting point, the pressure diﬀerence ∆p is adjusted so that the computed meniscus respects the prescribed contact angle θ1 at the component side (see the evolution of the contact angle θ1 as a function of ∆p for a given starting point in Figs. 14.5 and 14.6 and the evolution of the contact angle for several iterations of the starting point in Fig. 14.7). To do this, a starting pressure diﬀerence is chosen (dpstart ) and decremented by the quantity dpinc until the

128

14 Example 2: Numerical Implementation of the Proposed Models

INTPUTS

OUTPUTS

Literature models

Part and gripper material Surrounding environment

q1,q2 Volume repartition

SIMULATION TOOL

Liquid

γ

Volume of liquid Gripper geometry

Energy

Shape of the interface

Rupture distance Force exerted on the component

Pressure difference across the interface

Gap

Gripper kinematics

Newton's law

Component kinematics

Fig. 14.1. Overview of the inputs and outputs of the implemented model

−5

x 10

Gripper

8

z (m)

6

Starting point i+1 Meniscus

Starting point i Starting point 1

4 2 0 0.5

1

1.5 r (m)

2 x 10−4

Meniscus ij (starting point i, ∆ pj)

Fig. 14.2. The double iterative scheme for a spherical gripper (R = 0.1 mm), water, θ1 = θ2 = 30◦ , V = 4.5 nL, z = 0. Meniscus ij is obtained with the ith starting point and the jth pressure diﬀerence (Reprinted with permission from [108]. Copyright 2005 American Chemical Society)

θ D pi−1 D pi

qup θ

D p = f (θ)

q*

qlow

D pi+1 D p* D plow

Dp

ε

D pup

Fig. 14.3. θ as a function of ∆p: the interval ∆pup − ∆plow is divided into n subintervals

14.2 Liquid Bridge Simulation for the Analysis of a Meniscus

129

Choose a meniscus starting point Choose a pressure difference Contact angle OK?

No No

Volume OK? Compute the capillary force

Fig. 14.4. View of the resolution (Reprinted with permission from [108]. Copyright 2005 American Chemical Society) 100 90 80 70

θ1

60 50 40 30 20 10 0 −300

−250

−200

−150

−100

−50

0

dp

Fig. 14.5. First iterative scheme: ∆p (Pa) is adjusted to ﬁt θ1 (at this stage, the chosen volume has not been taken into account yet). Contact angle θ1 (◦ ) as a function of the pressure diﬀerence ∆p (Pa) across the interface: typical evolution (it is diﬃcult to adjust ∆p for small angles)

contact angle θ is bounded by θlow and θup given by two pressure diﬀerences ∆plow and ∆pup (Fig. 14.3): the quantity dpinc is modiﬁed if the sensitivity of θ is too small or too large. In some cases, if the angle is too large, the starting pressure diﬀerence (usually set around 10 Pa) is too small: it means that the pressure diﬀerence

130

14 Example 2: Numerical Implementation of the Proposed Models x 10−5 15

Starting Point

z [m]

10

5

0

−101

1

1.3

−201

1.35

−301 −401 −501 −601 −701 −801

1.4

1.45

r [m]

1.5 x 10−3

Fig. 14.6. Evolution of the meniscus proﬁle during the ﬁrst iteration scheme (the ﬁgures on right hand side of the curves indicate the pressure diﬀerence ∆p (Pa)) 1.06

Normalized angle

1.04 1.02 1 0.98 0.96 0.94

1

2

3

4 Iteration #

5

6

7

Fig. 14.7. Evolution of the normalized contact angle θ1 (i.e., computed θ1 divided by the prescribed one). The continuous lines state for the upper and lower imposed acceptable limits (±5%)

is actually positive (i.e., the pressure inside the meniscus is larger than the external pressure), leading to a meniscus about to collapse (or even nonexisting). Indeed, a positive pressure diﬀerence leads to a repulsive force that can be compensated by the attractive tension force component. It should also be emphasized that small contact angles can lead to instabilities because the

14.3 Evaluation of the Double Iterative Scheme x 10−3

4 Normalized volume

2

131

r2 [m]

1.5

1

3 2 1 0

0.5 1

2

3

4 5 Iteration #

(a)

6

7

2

3

4 5 Iteration #

6

7

(b)

Fig. 14.8. Evolution of the starting point and the computed volume for a spherical gripper (Φ = 8 mm), a gap z = 5 µm, water (γ = 72 × 10−3 N m−1 ), θ1 = θ2 = π3 , V = 100 nL. (a) Position of the iterated starting point (m); (b) Computed volume at each iteration

pressure diﬀerence switches from a value where θ1 exists but is too large to another value where θ1 does not exist, see Fig. 14.5. Once a pressure diﬀerence has been found to respect the contact angle, the volume associated to this meniscus is computed and compared with the prescribed one: if it is smaller (larger), the starting point is chosen farther from (nearer) the symmetry axis (see the typical evolution of the position r2 of the starting point in Fig. 14.8a and the corresponding computed volume in Fig. 14.8b). The normalized volume presented in Fig. 14.8b is the computed volume at each iteration divided by the prescribed volume: it oscillates until its value is inside the allowed error interval (represented by the two continuous lines, corresponding to an acceptable error ±5%). The ﬁrst iteration is not represented on this ﬁgure: the corresponding normalized volume is about 70.

14.3 Evaluation of the Double Iterative Scheme The inﬂuence of the number n of subintervals (see Fig. 14.3) in the determination of θ and ∆V V on the accuracy of the meniscus position have been investigated in the case of two parallel plates separated by a gap z = 50 µm. The volume of liquid (here water, i.e. γ = 72 × 10−3 N m−1 ) has been ﬁxed to 0.25 µL and the contact angles are both equal to π2 . In this case, the solution to the problem is a cylindrical meniscus whose radius R can be determined by expressing the volume: V = πR2 z

(14.1)

132

14 Example 2: Numerical Implementation of the Proposed Models

hence R = 1.262 mm. This result can be used to calculate the mean curvature H of the meniscus, since one curvature radius is given by R and the second one tends to ∞: 2H =

1 1 1 + = = 792.4 m−1 . R ∞ R

(14.2)

Consequently, by using (6.8), ∆p = 2Hγ = 57.07 Pa.

(14.3)

The results of this study are summarized in Tables 14.1 and 14.2 on the one hand and Figs. 14.9 and 14.10 on the other hand. In Table 14.2 and Fig. 14.10, the error on the radius Rneck is estimated as follows (the volume of liquid is assumed to be proportional to the cube of the characteristic size of the meniscus Rneck ): 3 αRneck =V

⇒ Rneck =

(14.4) V α

13 (14.5)

∆Rneck 1 ∆V = Rneck 3 V

(14.6)

Table 14.1. Inﬂuence of the discretization n of the interval [∆plow − ∆pup ] with dpinc = 1, dpstart = 100 Pa, ∆V /V = 1% n 5 10 20 50 100

∆p (Pa) 56 55.56 57.05 57.22 57.17

error (%) 14.29 6.40 2.95 1.14 0.57

Computation time (s) 29 38 49 94 168

Table 14.2. Inﬂuence of the relative error ∆V /V with dpinc = 1, dpstart = 100, n = 50 ∆V /V (%) 10 5 3.5 2 1 0.5 0.1

Rneck (mm) 1.250 1.250 1.250 1.250 1.260 1.260 1.261

∆Rneck /Rneck (%) 3.33 1.67 1.17 0.67 0.33 0.17 0.03

Computation time (s) 22 21 22 23 97 96 128

14.4 Pseudodynamic Simulation

133

60

∆p = 57.07Pa

50 40 30 20 10 0

Computing time [s]

Pressure difference [Pa]

70 150

100

50 # points

# points 0

20

40

60

80

100

0 0

120

(a)

20

40

60

80

100

(b)

Fig. 14.9. Inﬂuence of the discretization n of the interval [∆plow − ∆pup ] with dpinc = 1, dpstart = 100 Pa, ∆V /V = 1%.

150 R=1.26 mm Computing time [s]

Rneck [mm]

1.5

1

0.5

0 0

2

4

6

8

10

100

50

0

Relative error on the volume [%]

(a)

0

2 4 6 8 10 Relative error on the volume [%]

(b)

Fig. 14.10. Inﬂuence of the relative error ∆V /V with dpinc = 1, dpstart = 100, n = 50

14.4 Pseudodynamic Simulation To exploit the results in a dynamic simulation (for example, to predict if the omponent will stick to the gripper in an assembly machine or to design the kinematics of the gripper if it is intended to use its acceleration to release the component), we compute the position and velocity of the component at time step i+1 by solving the motion equation (Newton’s equation); the acceleration at time step i is deduced from the capillary force and the mass of the object. The position and the velocity of the object at time step i are used as boundary conditions. This procedure is summarized by the following equations: let us start with Newton’s law. m

d2 z = m¨ z = F. dt2

(14.7)

134

14 Example 2: Numerical Implementation of the Proposed Models

This second order equation is rewritten into a system of two ﬁrst-order equations: z Y ≡ , (14.8) z˙ Y (2, 1) Y˙ ≡ f (t, Y ) = . (14.9) F m

Hence, the Runge-Kutta method [49] allows to compute Yi+1 from Yi and Fi : 1 Yi+1 = Yi + (K1i + 2K2i + 2K3i + K4i ), (14.10) 6 where Yi (1, 1) is the position of the component at time step i and Yi (2, 1) is its velocity. As the force Fi can be computed at time step i from the meniscus geometry, the coeﬃcients Kji can be determined by K1i = ∆t.f (ti , Yi ), ∆t K1i , Yi + ), K2i = ∆t.f (ti + 2 2 ∆t K2i K3i = ∆t.f (ti + , Yi + ), 2 2 K4i = ∆t.f (ti + ∆t, Yi + K3i ).

(14.11) (14.12) (14.13) (14.14)

Figure 14.11 gives an overview of the complete algorithm. As far as the force computing is concerned, the most accurate results are found with the Time step i

Choose a meniscus starting point Choose a pressure difference No Contact angle OK? Volume OK?

No

Compute the capillary force

Compute object acceleration xi vi

Compute position and velocity at time step i+1

End simulation? Stop

Fig. 14.11. Algorithm of the coupled problem

No

14.5 Conclusions

135

“Laplace” calculation of the meniscus (=based on the Laplace equation), but for computing time reasons it is advised to use the “Arc” approximation (the force obtained by the “Arc” model is about a few % (2–5%) lower than the output by the Laplace model).

14.5 Conclusions This chapter has put the emphasize on the numerical implementation of the meniscus shape model based on the Laplace equation. From the knowledge of this meniscus shape, the capillary force acting in the component can be computed and the component moved according to the Newton’slaw. This implementation has been used to output the simulation results presented in Chap. 18. To experimentally validate these results, an experimental test bed has been set up, and this is explained in the third part of this book.

15 Conclusions of the Theoretical Study of Capillary Forces

In this second part of the book, we have introduced the underlying parameters (surface tension, advancing and receding contact angles) and models (Young– Dupr´e, Laplace, Cassie, Wenzel) ruling capillary forces. We have summarized approach and exact methods to compute these forces at equilibrium. For general meniscus shapes, the use of an energy minimization software such as Surface Evolver cannot be avoided, but we have shown how to solve the Laplace equation to compute the meniscus shape in axially symmetric cases, from which the force can be computed according to (6.11). This method has been applied successfully to the case study of a watch ball bearing, illustrating the existence of a force optimum in the case of a conforming gripper (i.e., a gripper whose geometry conforms with that of the component). The equivalence between this approach and the interfacial energy diﬀerentiation has been analytically proved in the case of a prism interacting with a plane. From the microassembly point of view, the ﬁrst conclusion is that the capillary forces linearly depend on the characteristic size of the meniscus; henceforth, they are of the utmost interest for miniaturization. Second, they provide a suitable principle to pick up ﬂat components whose only the top surface is accessible. Additionally, it has been shown how surface impurities or surface roughness theoretically aﬀect the contact angle. Nevertheless, the contact angle is considered as an input of the proposed models and will be measured in the characterization stage (see the third part of this book). Moreover, experiment will help to chose between advancing and receding contact angle. Since a useful release method in microassembly is to impose a high acceleration to the gripper, the suitability of an equilibrium modeling has been considered and it has been proved that it was an acceptable approach as long as the characteristic size of the meniscus would be smaller than a ¯ c , given by modiﬁed capillary length L γ ¯c = L (15.1) (a + g)ρ

138

15 Conclusions of the Theoretical Study of Capillary Forces

Consequently, we have not investigated the case of a dynamic simulation. Therefore, an exact estimation of characteristic times of this kind of picking was impossible, but an estimation of the characteristic damping time of the meniscus oscillations has been given on two diﬀerent ways, on the ﬁrst hand using dimensional analysis and on the other hand solving the Lucas–Washburn equation. An interesting damping phenomenon preventing high contact forces to damage the component has been pointed out from a ﬁgure of the literature (Fig. 11.5). Finally, the conditions for the rupture of the meniscus have been studied in the last chapter of this second part. Third part is devoted to experimental aspects, including both characterization to get the model input and measurement of the outputs.

16 Introduction

The third part of this book concerns the experimental aspects, which essentially covers three points, i.e., characterization, validation, and results. The characterization step consists in measuring the inputs of the models, that is, the contact angles, the surface tension, and the volume of liquid. The validation stage is the experimental veriﬁcation of the proposed models and simulations, mainly done by comparing two possible outputs, i.e., the capillary force and the meniscus shape. Thirdly, we present in this part a collection of results based on computations on the one hand and on experiments on the other hand. Therefore, this part falls into four chapters. Chapter 17 details the requirements list and the set up of the designed test bed, including the discussion of the possible sources of error. This test bed includes two main functions: vision (contact angles, gap, meniscus shape...) and force sensing. This chapter also summarizes the measurements of a ﬁrst characterization stage. The ﬁrst part of Chap. 18 contains the preliminary results that are used to validate the simulation code. It includes two meniscus shape validations, based on experiment and comparison with the analytical solution of the catenary curve (∆p = 0). The force output of the simulation is compared with an analytical benchmark and with experiments. The second part of Chap. 18 presents a detailed study on the inﬂuence of each parameter on the capillary force: the gap, the gripper geometry, the surface tension, the materials (through the contact angles). These results are compared with the simulation outputs. Then, some release concepts are tested: the inﬂuence of the relative orientation between the gripper and the component (the tilt angle breaks the axial symmetry assumption of the numerical model), the impact of an auxiliary PTFE tip, the use of a dynamical release strategy. The rupture distance of the meniscus is also experimentally studied in this chapter. The results of this chapter will be discussed in Chap. 21, especially the role of each parameter on the picking task and a quantiﬁed comparison of several release strategies,

142

16 Introduction

introducing the concept of adhesion ratio φ. These results are the ﬁrst steps toward design rules of a surface tension based gripper. In the two last chapters, these concepts are then applied to the case study of a watch ball bearing. Following an identical methodology, Chap. 19 concerns the characterization aspects (grippers geometry, contact angles, surface tension) while Chap. 20 includes the results obtained with the designed microgripper. This chapter presents the studies of the picking task (with the possible errors and some solutions) and the releasing task (including a study on the reliability of this principle). The force model presented in Chap. 10 is experimentally assessed. The force order of magnitude given by this model is validated but a more accurate validation would require the exact determination of the involved volume of liquid. This is also exhaustively discussed in Sect. 20.5. The conclusions proposed in this part put forward some design rules for surface tension based grippers.

17 Test Bed and Characterization

17.1 Introduction The need for a test bed can be justiﬁed by several reasons: the main one is probably the need for an experimental validation of the above described simulation. Moreover, the inherent assumptions of the built model prevent us from getting results from nonaxially symmetric conﬁgurations. A test bed can then feed us with additional information such as, for example, the inﬂuence of the gripper tilt on the capillary forces. A third reason is that the simulation has to be fed with input data such as the amount of liquid and the contact angles: these inputs will be measured with this test bed. Moreover, it is still not clear which contact angle (advancing one vs receding one) should be used: the choice will be achieved by direct observation of the picking operation. Consequently, this chapter focuses on the requirement and the design principles of a force measurement test bed, which can be sketched as shown in Fig. 17.1. The information released in this chapter concerns the integration of each function (force measurement, position sensing, liquid dispensing, vision), the calibration and characterization of the test bed itself, and ﬁnally, the characterization of the materials involved in the experiments and the related contact angles.

17.2 Requirements In Table 17.1, X axis refers to the optical axis of the camera (see Fig. 17.1) and Z axis to the “vertical”1 axis, i.e., the symmetry axis of the gripper.

1

The term “vertical” usually refers to gravity, but here gravity is neglected so that we use it to refer to the symmetry axis of the gripper.

144

17 Test Bed and Characterization Positioning sensor Upper solid (gripper)

d2

Liquid dispensing

Z

(Compliance)

Meniscus

d1

Imaged scene Force sensor

Y X

(Deflection sensor)

Fig. 17.1. Principle of the force test bed: the variation of d1 is measured by a noncontact displacement sensor. This variation, together with the stiﬀness of the cantilevered beam, gives the amount of force applied onto this beam. This force is exerted by a meniscus that has been dispensed between the beam, which acts as a component, and the upper solid, which acts as a gripper. The position d2 of the upper solid is also measured and the knowledge of both d1 and d2 gives the gap between the upper solid and the beam. Finally, the stripped box indicates the imaged scene.

Table 17.1. Requirements for the test bed Category Requirement Force Measurement Measurement along the Z axis Full scale: several milli-newton Sensitivity: several micro-newton Vision Vision ﬁeld: about 1 mm2 Range of focusing distance: several millimeter (X) 2 dof for the camera in the plane Y Z Zoom ×40 Image acquisition and transfer to the work station Backlight system with its power supply Gripper interface 2 dof along X and Y , low accuracy 1 dof along Z axis, high resolution (≈ 1µm), range ≈ 5 mm Interchangeable gripper interface 1 rotation dof (θx ) to tune the tilt of the gripper (range ≈ 45◦ , accuracy better than 1◦ ) Components Flat components, several materials Drop delivery Water and silicone oil delivery Volume smaller than 1µL Displacement 2 mm range along Z (to induce the bridge collapse) measurements 1 µm accuracy

17.3 Test Bed Principles

145

17.3 Test Bed Principles 17.3.1 Force Measurement According to Ref. [67], the sensing of small forces can be achieved by measuring the displacement caused by elastic deformation induced by the loading of an elastic structure. Still according to Ref. [67], usual deformable sampling bodies are presented in Fig. 17.2a. The most suitable body is the cantilevered beam (Fig. 17.2b) because it can serve both as deformable body for the force sensing and as component for the experiment. If the component material has to be changed, another blade can be used or a small ﬂat sample can be glued on it. Because of its small thickness, the blade extremity undergoes a deﬂection δweight due to its own weight [150]:

δweight =

3 ρg L4 qL4 = 8EI 2 E h2

(17.1)

where L is the length of the beam (m), q is the distributed load per unit length (N m−1 ), i.e., q = ρghb, where ρ is the density of the beam (kg m−3 ), g is the well-known gravity constant (9.81 ms−2 ), h is the beam thickness (m), and b is the beam width (m). E denotes the Young modulus of the material (Pa) and I refers to the momentum of the beam section, i.e., I = bh3 /12 (m4 ). Typical values are given in Tables 17.2 and 17.3. Note that this deﬂection does not actually disturb the measurement because deﬂections are assumed to be small. Consequently, the linearity and the superposition principle can be applied and the own deﬂection neglected.

F

δ

(a)

(b)

Fig. 17.2. Force sensing (a) deformable sampling bodies [67]; (b) chosen force sensing principle Table 17.2. Geometrical properties of the cantilevered blades Symbol L h b

Description Length Thickness Width

Value 10–100 0.025–1 12

Units 10−3 m 10−3 m 10−3 m

146

17 Test Bed and Characterization

Table 17.3. Physical properties of the cantilevered blades Symbol E ρ

Description Young modulus Density

Steel 210 7,800

Units 109 Pa kg m−3

Actually, in (17.1), δweight cannot directly be reduced because it should imply a loss of sensitivity as far as the force measure is concerned. Indeed, the deﬂection δforce caused by an external load F (expressed in N) is given in [150]: δforce (L) =

4F L3 F L3 = . 3EI Ebh3

(17.2)

Consequently, it should be tried to maximize the following ratio M , describing the relative importance due to force deﬂection compared to that due to weight deﬂection:

M=

8F 1 δforce . = δweight 3ρgb hL

(17.3)

If the density of the cantilever and its width are ﬁxed, we have the following two considerations for a given force: 1. The thickness h and the length L should be as small as possible in order to reduce the relative importance of the weight 2. h should be as small as possible in order to maximize the force measurement sensitivity, but the length L should be maximized We conclude that the thinnest cantilever should be chosen, and its length adapted for minimal sensitivity. The deﬂection measuring system has been implemented as follows: at equilibrium the capillary upward force F exerted by the liquid bridge on a cantilevered beam is balanced by the elastic restoring force due to the beam deﬂection δ (Fig. 17.2b). This deﬂection is directly measured by a noncontact displacement sensor (Keyence LC-2440 laser), whose measuring range is equal to 3 mm and accuracy is guaranteed by the constructor to be 0.2 µm. The measured value can be read immediately on the controller display or can be transmitted by a RS-232 connection in order to achieve almost real time acquisition. 17.3.2 Drop Dispensing One must distinguish between continuous ﬂow and drop-on-demand generators. Basically, the main parameters governing the drop-on-demand delivering are surface tension, viscosity, inner diameter of the ejection nozzle, ejection pressure [19]. Several principles of drop-on-demand systems are described in

17.3 Test Bed Principles

147

[116]. The commercially available systems are compared in Table 17.4 (Microdrop and Gesim companies). Drop-on-demand generators are quite expensive but allow to generate smaller drops than the manual dispensing device. Nevertheless, it is not justiﬁed because the drop size range of the manual dispensing device (Eppendorf company) is small enough to meet the assumptions of the model to be validated: a droplet diameter between 500 and 1,000 µm is smaller than the capillary length (LC ≈ 2.5 mm). Moreover, the size that must be smaller than the capillary length is actually the droplet height and not its diameter, which is the case in near contact conﬁgurations. A drawback of both systems (i.e., manual dispensing systems and automated drop-on-demand generators) is that they are calibrated for water: the drop-on-demand generators can also dispense liquids like silicone oil but, in this case, they require an additional heating unit to decrease the liquid viscosity. For these experimental tests, a manual dispensing system has been used. Since the liquid amount is an input parameter of the simulation tool, it has to be known with accuracy. It is either calibrated with the manual dispensing device (from 0.1 to 2.5 µL, with steps equal to 2 nL) or it can be measured from the height h and the diameter D of the droplet as depicted in Fig. 17.3. Indeed, the radius R and the cosine of the angle cos θ can be expressed starting from R(1 − cos θ) = h, 2R sin θ = D, ⇒2

(17.4) (17.5)

h sin θ + cos θ = 1, D

(17.6)

Table 17.4. Comparison between commercial solutions for drop generation Permissible liquid viscosity (mPa s) Inner nozzle diameter µm Dosing volume (nL) Drop diameter (µm) Drop range of ﬂight (mm) Price (e) Actuation

Microdrop 10 30–70 0.048–0.144 45–65 20 6,300 Piezo

Gesim 5 ? 0.1–1 58–124 1 4,470 Piezo

h

θ D Fig. 17.3. Volume measurement from a spherical cap picture

Eppendorf water 100–2,500 576–1,684 Contact 200 Manual

148

17 Test Bed and Characterization

which leads to D2 + 4h2 , 8h D2 − 4h2 cos θ = 2 , D + 4h2 hence the volume can be determined: R=

V =

3 1 2πR3 (1 − cos θ + cos3 θ). 3 2 2

(17.7) (17.8)

(17.9)

Let us note that this second method is not suitable for very small contact angles, because the height h becomes too small to be measured accurately from a number of pixels. 17.3.3 Vision The used camera is the Keyence-CV050 CCD camera, mounted with a 50 mm lens and a set of rings to tune the zoom. This device is coupled to a monitor allowing to track the picture, to achieve several measurements,2 and to transfer the acquired image to the work station. For imaging droplets, a backlight illumination system is recommended.

17.4 CAD Model and Drawings CAD drawings of the designed set up have been made with CATIA (V5) and are presented in Fig. 17.4, where the following elements can be seen: the beam is located just below the spherical tip stating for the gripper. Its extremity is located in the view ﬁeld of the camera and its cantilevered length can be changed by modifying the casing to tune ﬂexibility. The camera is mounted on a plate that can be moved along the X axis and in the plane Y Z with two manual stages. The gripper tip (here it is a spherical tip) can be changed easily, thanks to the gripper interface. This one is mounted on a manual stage allowing a relative motion along the Z axis with respect to the upper displacement sensor. Both can be moved together in the XY plane, thanks to a manual stage mounted on the top of a gantry. The role of the latter is to carry a back light represented in Fig. 17.5 that is used to improve the illumination conditions. These drawings can be compared to the pictures of the set up shown in Figs. 17.5, 17.6, and 17.7.

2

Contact angle measurement, gripper geometry, volume of liquid.

17.4 CAD Model and Drawings

Displacement sensors

149

Z

Spherical Tip

Z

270 mm

Gripper Interface

1D manual stages

Beam

XY

Gantry

Camera

X

Y

Casing YZ

X YZ

130 mm 2D manual stages

(a)

(b)

Fig. 17.4. Drawings of the experimental set up (a) left side view; (b) right side view (Reprinted with permission from [108]. Copyright 2005 American Chemical Society)

Displacement sensor Z axis

Gripper interface XY X axis

Spherical tip Camera

Y axis

Back light

Fig. 17.5. Left side view (Reprinted with permission from [108]. Copyright 2005 American Chemical Society)

150

17 Test Bed and Characterization

40mm

(a)

(b)

Fig. 17.6. Experimental set up (a) front view; (b) right side view

(a)

(b)

Fig. 17.7. Experimental set up (a) side view; (b) detailed view

17.5 Characteristics of the Force Measurement Set Up

151

17.5 Characteristics of the Force Measurement Set Up 17.5.1 Typical Calibration The typical way to calibrate the set up consists in measuring the deﬂection δ due to a known force, such as, for example, the weight of a calibrated mass m0 . Typical values are the following: • •

m0 = 301.6 mg ± 0.1 δ = 510 µm ± 20

The mass has been weighted with both a mechanical and an electronic balance, with a 0.1 mg accuracy. The accuracy of the deﬂection measure lies more in the fact that there is an error in the positioning of the calibrating mass on the beam, leading to an error on the deﬂection that has been estimated at about 20 µm by estimating the extreme possible positions for the calibrating mass. Consequently, the stiﬀness of the beam is given by m0 g = 5.80 N m−1 , (17.10) k= δ with a relative error given by ∆k ∆m0 ∆δ 20 ∆δ = ≈ ≈ ≈ 4%. + k m0 δ δ 580

(17.11)

This result can be improved by calibrating the system with a heavier mass, so that δ is increased and that the error is decreased. Nevertheless, the deﬂection cannot be increased above 2–3 mm so that the best accuracy could be ∆δ 20 ∆k ≈ ≈ ≈ 1%. k δ 2000

(17.12)

17.5.2 Linearity To assess the linearity of the set up, Fig. 17.8 presents the measured deﬂection as a function of the applied force, i.e., the weight of reference masses (masses mi and deﬂections δi are presented in Table 17.5). If δi is supposed to be approached by δi ≈ ami + b where a and b are obtained by the least square method: N Σ(δi mi ) − Σδi Σmi , N Σm2i − (Σmi )2 Σδi Σm2i − Σ(δi mi )Σmi , b= N Σm2i − (Σmi )2

a=

(17.13) (17.14)

the linearity error is deﬁned in [67] as ≡

max |ami + b − δi | . max δi

(17.15)

Since a = 1.7144 µm g−1 and b = 0.735 µm, the linearity error is about 0.33%.

152

17 Test Bed and Characterization

Measured Deflection [m]

1.5

x 10−3

1

0.5

0

0

0.002

0.004

0.006

0.008

0.01

Applied Force [N]

Fig. 17.8. Linearity of the force measurement set up Table 17.5. Data for the linearity mi (mass in g) 0 0.1225 0.2453 0.3643 0.4836 0.6049 0.7490 0.8608 δi (deﬂection in µm) 0 210 423 627 832 1034 1280 1481

17.5.3 Accuracy As the measured force is given by F = kδ,

(17.16)

the measurement error is given by ∆F ∆k ∆δ = + . F k δ

(17.17)

The relative error on the stiﬀness has been estimated at about 1–4% and the absolute error on the deﬂection about 20 µm. If the stiﬀness of the beam is adjusted (by changing its length for example) in order to get deﬂection about 600–2,000 µm, the measurement error of the force can be estimated at: 20 ∆F ≈ 1 − 4% + ≈ 1 − 4% + 1 − 3.3% ≈ 2 − 7.3%. F 600 − 2000

(17.18)

17.5.4 Inﬂuence of a Misalignment on the Force Measurement It must be distinguished between the misalignment of the gripper and the component on the one hand (Fig. 17.9) and the misalignment between the gripper and the droplet on the other hand (Fig. 17.10). 1. Misalignment of the gripper and the component: when there is no alignment error between the component and the gripper (it is the case if their

17.5 Characteristics of the Force Measurement Set Up z

z

O F

P

z'

ε

Spherical gripper φ

Spherical component

Spherical component

P

Beam

Fmeas

153

F O

Q

Spherical gripper

Beam

Fmeas

(a)

(b)

Fig. 17.9. Centering error between a spherical component and a spherical gripper. (a) Without centering error; (b) with a centering error

(a)

(b)

(c)

(d)

Fig. 17.10. Centering error between the delivered droplet and a spherical gripper

centers P and O are on the same vertical z axis, see Fig. 17.9a), the measured force Fmeas (actually the restoring elastic force of the deﬂected beam) is equal to the capillary force F between the component and the gripper. Let us now assume that the gripper is not perfectly aligned with the component and that there is a distance between the z axis of the component and the z axis parallel to z and containing the center O of the gripper (Fig. 17.9b): in this case Fmeas is equal to the projection of F : Fmeas = F cos φ = F 1 − sin2 φ 2 = F 1− R1 + R2 2 1 ≈ F 1− 2 R1 + R2

(17.19)

where φ can be read in Fig. 17.9b and R1 and R2 state for the component and gripper radii. is assumed to be always smaller than 0.5 mm and R1 + R2 ≈ 5 mm. Then the ratio Fmeas /F is about 99.5%. Consequently, this error will be neglected. 2. Misalignment between the gripper and the droplet: this misalignment (Fig. 17.10a) does not imply measurement error because once the liquid bridge has linked the gripper and the component (Fig. 17.10b), there is

154

17 Test Bed and Characterization

a centering eﬀect (Fig. 17.10c) of the liquid that makes the alignment perfectly correct after the ﬁrst rupture (Fig. 17.10d). Note that measurements have always been preceded by a blank trial.

17.6 Characteristics of the Contact Angles Measurements As the contact angles constitute inputs for the simulations, it was necessary to measure them according to the diﬀerent solid–liquid combinations (for example, steel–water, steel–silicone oil). The ﬁrst idea was to assume that a small droplet posed on a substrate would take a spherical cap shape (see Fig. 17.11a), the contact angle θ can consequently be deduced from the drop height h and the drop diameter D, measured on the screen of the CCD camera controller. Using (17.6) and assuming t ≡ tan θ/2, sin θ = 2t/1 + t2 and cos θ = 1 − t2 /1 + t2 , we get tan

θ 2h =t= . 2 D

(17.20)

Unfortunately, as already mentioned, this method does not hold when θ is small because in this case it is diﬃcult to determine h, accurately, and when the hysteresis is large because it is diﬃcult to know whether the angle made by the spherical cap tends or not to the advancing or receding contact angles. Consequently, the ﬁnally used method consists in transforming the droplet in a meniscus by approaching the gripper near the blade on top of which the drop is posed. Once the meniscus has been formed, the gripper is slightly moved downwards (upwards) to force the contact line to move along the solids: in this case we can be ensured to measure the advancing (receding) contact angle as depicted in Fig. 17.11b. The contact angle is then determined by the angle between the line tangent to the meniscus and the one tangent to the solid.

D

θ

h

θ

R qR

qA qA

(a)

(b)

qR

(c)

Fig. 17.11. Diﬀerent measured contact angles. (a) Spherical cap – not well deﬁned angle; (b) advancing contact angle; (c) receding contact angle

17.8 Modus Operandi

155

The results presented in Table 17.9 show a dispersion of the angle measurements, which is quite moderate for silicone oil but which can be dramatically high for water. Hopefully, the pertinent information seems to be cos θ more than θ so that even with ∆θ ≈ 15◦ , the error on the cosine remains lower than 7%. Additional information as far as the contact angles are concerned can be found in Sect. 18.3.

17.7 Surface Tension Measurement If necessary, the surface tension can be measured, thanks to classical methods described in [1], such as, for example, the Wilhelmy plate method or the Du Nou¨ y ring method. In this case, surface tension has not been measured directly since the liquid which have been used in this part of the work are well characterized ones, whose properties are given in Sect. 17.9.4.

17.8 Modus Operandi 1. The gripper and the beam are chosen and set up: a small ﬂat component made of whatever material that can be glued onto the beam, in order to simulate the desired material. Note that this operation does not disturb the force measurement as the weight of the additional component is quite small. 2. The force measurement system is calibrated by measuring the deﬂection corresponding to a known force, i.e., the weight of a reference mass (Fig. 17.12). 3. The gripper is moved downwards until the contact with the beam is detected (Fig. 17.13).

(a)

(b)

Fig. 17.12. Test bench calibration. (a) Calibration mass; (b) corresponding deﬂection

156

17 Test Bed and Characterization

(a)

(b)

Fig. 17.13. Initial conﬁguration of the gripper and the beam. (a) Away from contact (beam width = 12.7 mm); (b) at contact

(a)

(b)

Fig. 17.14. 0.5 µL water droplet. (a) Droplet, beam, and spherical tip gripper (beam width = 12.7 mm); (b) detailed view

4. Displacement sensors are set to zero. 5. The gripper is moved upwards to free space between its tip and the beam. A liquid droplet is then put on the beam right below the gripper tip (Fig. 17.14a): its volume can be known either by using the calibrated indication of the micro-pipetting device or by computing the volume from the geometrical parameters of the spherical cap-shaped droplet shown in Fig. 17.14b. A last solution consists in superposing the meniscus proﬁle and the computed shape for a given input volume. If the shapes correspond with one another, then the volumes are equal. 6. The gripper is moved downwards until the droplet turns itself into a liquid bridge. From this situation, the advancing (receding) contact angles are obtained by slowly moving the gripper downwards (upwards) and measured directly on the screen of the camera monitor (Fig. 17.15). Results are presented in Table 17.9.

17.8 Modus Operandi

(a)

157

(b)

Fig. 17.15. Steel component (St-004-02), water, spherical gripper (Φ = 13 mm). (a) Advancing contact angle; (b) receding contact angle

Fig. 17.16. Deﬂection measurements (mm)

7. Force measurement: the separation distance can be tuned by moving the gripper. The cantilever deﬂection sensor should now display a positive value because the cantilevered blade is pulled upwards by the meniscus, while the gripper sensor should display a negative value because the gripper is on a higher position than at the initialization time (see Fig. 17.16). The sum of the values leads to the separation distance. a) The force at contact can be measured from the maximum deﬂection of the beam. Note that this maximum can be diﬃcult to read because this conﬁguration is unstable: indeed, from the maximal deﬂection situation, the beam quickly jumps downwards when the gripper height increases. The reason therefore is that the capillary force becomes lower than the elastic restoring force of the beam, and consequently, there is a jump from this position to a lower one, corresponding to another forces balance, given by a larger gap. According to the stiﬀness of the beam, this second equilibrium position can exist or not (this phenomenon is widely described in [31]).

158

17 Test Bed and Characterization

Fig. 17.17. Example of calibration with a ceramic slip gage (width = 1 mm)

b) Force–distance curve can be drawn by moving the gripper downwards step by step until the physical contact of the gripper and the cantilever: during this phase, the deﬂection of the cantilever gives the value of the pulling force exerted by the meniscus. Note that in step 5, it can be necessary to calibrate the camera: this is achieved by imaging a ceramic slip gage (width = 1 mm), as depicted in Fig. 17.17. We then conclude that the scale is about 330 ± 2 pixel mm−1 .

17.9 Characterization 17.9.1 Set of Available Grippers First let us note that gripper here must be understood as the upper solid. The case of an actual gripper will be considered in Chap. 19. As we wanted to study the inﬂuence of the shape (both the gripper type and the gripper parameter p), conical and spherical grippers have been considered. •

The conical grippers have been turned in steel material: – GC-St-0: cylindrical gripper p = 0◦ (polished, Ra = 0.23 ± 0.01 µm) – GC-St-5: conical aperture angle p = 4.5◦ , Ra = 0.23 ± 0.01 µm (Fig. 17.18a) – GC-St-10: conical aperture angle p = 9.75◦ , Ra = 0.23 ± 0.01 µm (Fig. 17.18b) – GC-St-45: conical aperture angle p = 47.5◦ , Ra = 0.23 ± 0.01 µm (Fig. 17.18c)

17.9 Characterization

(a)

(b)

159

(c)

Fig. 17.18. Geometry of the conical tips (a) GC-St-5; (b) GC-St-10; (c) GC-St-45

(a)

(b)

(c)

Fig. 17.19. Geometry of the spherical tips. (a) GS-St-3.2; (b) GS-St-7.9; (c) GSSt-13.0

•

•

The spherical grippers are made with steel ball bearings (the roughness Ra has not been measured but values are given according to ISO3290:1998 found in [42], p. 595): – GS-St-3.2: Φ3.2 mm (Ra < 0.010 µm) (Fig. 17.19a) – GS-St-7.9: Φ7.9 mm (Ra < 0.010 µm) (Fig. 17.19b) – GS-St-12.8: Φ12.8 mm (Ra < 0.014 µm) – GS-St-13: Φ13.0 mm (Ra < 0.014 µm) (Fig. 17.19c) Finally a foam gripper (2×2.5×2 mm3 ) has been tested in dynamic release trials

17.9.2 Set of Available Components The steel components have been cut in a steel blade with a thickness 102 ± 5 µm (accuracy referred by the supplier Precision Brand). The length and the width have been roughly measured because of the shape errors of the components, the mass with a 0.1 mg accuracy weighting balance. The roughness has been measured with a Taylor Hobson device (±0.01 µm): two values are given (when measured) parallel and perpendicular to the machining direction of the blade (see Tables 17.6 and 17.7).

160

17 Test Bed and Characterization

Table 17.6. Steel components Code Material Thickness (µm) Length (mm) Width (mm) ±5 St-1 Steel 102 3.00 3.30 St-2 Steel 102 5.50 5.60 St-3 Steel 102 7.55 8.30 St-4 Steel 102 12.70 15.00

Ra (µm) Mass (g) ±0.01 ±0.0001 0.09–0.23 0.0072 0.09–0.23 0.0228 0.09–0.23 0.0469 0.09–0.23 0.1428

Table 17.7. Silicon components Code Material Thickness (µm) Length (mm) Width (mm) Ra (µm) Mass (g) ±50 ±0.01 ±0.0001 Si-1 Silicon 550 3.30 4.00 <0.03 0.0128 Si-2 Silicon 550 5.75 6.45 <0.03 0.0385 Si-3 Silicon 550 5.75 8.15 <0.03 0.0508 Si-4 Silicon 550 11.85 13.60 <0.03 0.1827 Si-5 Silicon 550 0.1550 Si-6 Silicon 550 0.1713 Si-7 Silicon 550 0.2850 Si-8 Silicon 550 0.2023

Table 17.8. Blades Code St-004-01 St-004-02 St-004-04 Si-004-01 Si-004-02

Material Steel Steel Steel Silicon Silicon

Thickness (µm) ±5 102 102 102 102 + (550 ± 50) 102 + (550 ± 50)

Width (mm) ±0.13 12.70 12.70 12.70 12.70 (without object) 12.70 (without object)

Ra (µm) ±0.01 0.23 0.10–0.29 Not measured <0.030 <0.030

The silicon components have been cleaved in a silicon wafer with a thickness 550 ± 50 µm. 17.9.3 Set of Available Blades The basic blade consists in a 102 µm thick steel blade (St-004-XX). The blades Si-004-XX consist in silicon ﬂat components glued on the steel blade. In this case the dimensions of the silicon components are indicated next to the dimensions of the steel blade in Table 17.8. The width is given with the supplier accuracy (±0.13 mm). Again two values for Ra are given parallel and perpendicular to the machining directions.

17.9 Characterization

(a)

161

(b)

Fig. 17.20. Steel component (St-004-02), water, conical gripper (GC-St-10). (a) Advancing angle; (b) receding angle

17.9.4 Available Liquids Two liquids have been used: 1. Water (milli-Q purity grade), ρ = 1, 000 kg m−3 , γ = 72 × 10−3 N m−1 , µ25◦ = 1 × 10−3 Pa s 2. R47V50 silicone oil, ρ = 960 kg m−3 , γ = 20.8 × 10−3 N m−1 , µ25◦ = 48 × 10−3 Pa s 17.9.5 Contact Angles Characterization The way the static contact angles have been determined has already been brieﬂy presented in Sect. 17.6. For each conﬁguration (=a given liquid and a given solid) the meniscus has been imaged ﬁve times for advancing and receding contact angle (see for example Fig. 17.20). This was achieved with an almost zero (a few µm s−1 ) velocity and so the measured angles correspond to the advancing and receding static contact angles. For each image the angle has been measured three or four times: the results are presented in Table 17.9. In this table, liquid I was water and liquid II was silicone oil (R47V50). The solids that have been tested are representative for all grippers and components intended to be used later: they have been chosen either because of the constitutive material (steel, silicon) or because of their surface roughness. Consequently, the GS-St-13.0 spherical gripper made of a steel ball (for ball bearings) represents all spherical grippers. With the same idea, the GC-St-10 states for all conical grippers. The two tested components were steel (St-000402) and silicon (Si-004-01) blades. The measured angles have been averaged (θA and θR ) and their standard deviation computed (σθA and σθR ). The relative errors on the cosines have been computed from measures that were inside the intervals θ ± 2σ and θ ± 1.5σ. Finally, the hysteresis has been computed as the diﬀerence between the mean advancing contact angle θA and the mean receding contact angle θR .

162

17 Test Bed and Characterization

Table 17.9. Advancing and receding contact angles Liquid

Solid

H2 0(I) H2 0(I) H2 0(I) H2 0(I) H2 0(I) Oil(II) Oil(II) Oil(II) Oil(II) Oil(II)

St-004-02 Si-004-01 GS-St-13.0 GC-St-10 GC-St-00 St-004-02 Si-004-01 GS-St-13.0 GC-St-10 GC-St-00

θA (◦ ) 92 55 88 82 92 34 29 22 23 33

θR (◦ ) 56 33 37 25 39 16 16 12 13 14

σθA (◦ ) 1.9 6.0 8.1 12.9 2.4 7.1 9.5 5.6 6.0 7.2

cos θA ∆ cos θR ∆ cos θA ∆ cos θR σθR ∆cos Hysteresis θA cos θR cos θA cos θR ◦ ( ) (1.5σθa ) (1.5σθr ) (2σθa ) (2σθr ) (◦ ) 8.3 1.39 0.32 1.86 0.42 36 4.6 0.23 0.08 0.30 0.10 23 4.4 4.88 0.09 6.51 0.11 51 7.9 2.53 0.10 3.37 0.13 57 15.6 2.10 0.33 2.81 0.44 53 3.5 0.13 0.03 0.17 0.04 18 5.8 0.14 0.04 0.18 0.06 13 3.3 0.06 0.02 0.08 0.02 10 2.4 0.07 0.01 0.09 0.02 11 2.8 0.12 0.02 0.16 0.03 18

Additional tests have been led with PTFE. The receding contact angle with R47V50 was [44, 46, 50]◦ → 47◦ , that one with water was [100, 99, 94]◦ → 96◦ .

17.10 Conclusions This chapter presented the design of the test bench, its performances and error sources, and the related modus operandi. Then the diﬀerent grippers, components, and blades were described and characterized. This equipment has been used to measure the static contact angles and the contact angle hysteresis, which will be used as inputs of the models in the next chapter.

18 Results

18.1 Introduction Two groups of results are presented in this chapter. Section 18.2 summarizes preliminary experiments, which have been led in order to validate the simulation tool: the corresponding results concern (1) the meniscus proﬁle, (2) the comparison between the simulation outputs and analytical results, and (3) the comparison between the simulation outputs and experimental results. Sections 18.3–18.12 collect both simulation and experimental results. The results obtained in relation with the watch bearing case study already presented in Chap. 10 will be given in Chaps. 19 and 20.

18.2 Preliminary Results: Validation of the Simulation Code To validate the developed simulation code, we have compared its results as far as both the meniscus shape and the capillary force are concerned. The meniscus shape has been studied in two simpliﬁed cases and a comparison has been led with experiment in a general case. Then, we have compared the output force both with the approximations of the capillary force from the literature and with experimental results. 18.2.1 Meniscus Proﬁle The ﬁrst meniscus proﬁle validation is the case of two parallel plates separated by a distance b and for a diﬀerence of pressure equal to zero, leading to the analytical equation of a catenary curve (see Appendix B.3): r(z) = A cosh

z−B A

(18.1)

164

18 Results x 10−3 2

z [m]

1.5

1 Analytical solution Numerical solution

0.5

0

0

0.5

1

1.5

2

2.5

r [m]

3 x 10−3

Fig. 18.1. Comparison between the numerical and analytical meniscus shape, with ∆p = 0, a = 3 mm, b = 2 mm, and θ2 = 60◦ (Reprinted with permission from [108]. Copyright 2005 American Chemical Society.)

where A = a sin θ and B = b − a sin θacosh(1/sin θ) (a, b, and θ are shown in Fig. 18.1). For z = 0, the relative error between the numerical radius and the analytical one is about 1.5%. A second case has been tested, namely the case of a meniscus between two parallel plates, with contact angles equal to 90◦ , leading to a cylindrical meniscus (with radius R) whose principal curvature radii are R1 = ∞ and R2 = R. This case has already been presented in Sect. 14.3: the graphical output of the meniscus shape is presented in Fig. 18.2a. The last veriﬁcation operated on the meniscus shape is the comparison between the output proﬁle and the picture of the meniscus, as presented in Fig. 18.2b. This picture corresponds to a 12.7 mm diameter spherical steel gripper catching a steel component (blade St-004-2) with 0.72 µL water. The gap in this case is 265 µm. The dashed line states for the meniscus output by the simulation tool. 18.2.2 Comparison with the Analytical Expressions Let us remind the analytical approximation of the capillary force between a plane and a sphere (radius R), for a gap equal to zero and an equivalent θ2 ). As usual γ is the surface contact angle θ (if θ1 = θ2 , cos θ ≡ cos θ1 +cos 2 tension of the liquid between the gripper and the component. F = 4πRγ cos θ.

(18.2)

18.2 Preliminary Results: Validation of the Simulation Code

165

z [m]

x 10−4 6 5 4 3 2 1 0 −1 −2 −3

Measured interface Liquid bridge

z=265µm

Simulation,V=0.72µL

0

0.2

0.4

0.6 0.8 r [m]

1

1.2 x 10−3

(a)

(b)

Fig. 18.2. Study of the meniscus shape (a) Simulation: meniscus shape with θ1 = θ2 = 90◦ (V = 0.25 µL, z = 50 µm); (b) Comparison between the simulation and the experimental meniscus shape (water, steel component, spherical steel gripper (R = 6.35 mm)) (Both reprinted with permission from [108]. Copyright 2005 American Chemical Society)

V=0.1µl (water) Analytical (water) V=0.1µl (R47V50) Analytical (R47V50) V=0.5µl (water) V=0.5µl (R47V50) V=1µl (water) V=1µl (R47V50)

0.012 0.01

Force [N]

0.008 0.006 0.004 0.002 0 10

20

30 40 50 Contact angles θ1 = θ2 [8]

60

70

Fig. 18.3. Comparison between the simulation results and the analytical approximation F = 4πγR cos θ for a spherical gripper with diameter 13.0 mm, γ = 72 × 10−3 N m−1 (water) and γ = 20.8 × 10−3 N m−1 (R47V50). The results are presented for diﬀerent volumes (0.1, 0.5, and 1 µL) and diﬀerent contact angles simulating diﬀerent materials; the simulation points tend to the analytical approximations for water (solid line) and silicone oil (dashed line) (Reprinted with permission from [108]. Copyright 2005 American Chemical Society)

166

18 Results

Figure 18.3 plots the force as a function of the equivalent contact angle θ for a Φ 26 mm spherical gripper and for two liquids: the upper curve shows the force for water and the lower one that for silicone oil (R47V50). It can be seen on this picture that the results of the simulations tend to the analytical approximations (solid and dashed lines). 18.2.3 Experimental Validation Several experimental validations have been led: the tested conﬁgurations are summarized in Table 18.1. Each experiment has been made 15 times, and each time plotted in Figs. 18.4 and 18.5 by a dot. The “cross” signs state for the boundaries of the error interval, centered on the mean value of the 15 results. This error interval has been calculated as indicated in Sect. 17.5.3: ∆k ∆δ ∆me g ∆δe 20 µm 10 µm ∆F ∆δ = + = + ≈0+ + (18.3) + F k δ me g δe δ 600 µm min(δi ) Table 18.1. Summary of the experimentally tested conﬁgurations: 1 refers to components and 2 refers to steel grippers, A refers to advancing and R refers to receding. The angles values shown in this table have been separately measured (see Table 17.9) # Gripper radius Component Liquid θ1R θ2R θ1A θ2A γ (mm) (◦ ) (◦ ) (◦ ) (◦ ) (mN m−1 ) 1 6.4 Steel R47V50 16 12 34 22 20.8 2 6.4 Steel R47V50 16 12 34 22 20.8 3 6.4 Steel R47V50 16 12 34 22 20.8 4 1.6 Steel R47V50 16 12 34 22 20.8 5 1.6 Steel R47V50 16 12 34 22 20.8 6 1.6 Steel R47V50 16 12 34 22 20.8 7 1.6 Silicon R47V50 16 12 29 22 20.8 8 1.6 Silicon R47V50 16 12 29 22 20.8 9 1.6 Silicon R47V50 16 12 29 22 20.8 10 6.4 Silicon R47V50 16 12 29 22 20.8 11 6.4 Silicon R47V50 16 12 29 22 20.8 12 6.4 Silicon R47V50 16 12 29 22 20.8 13 1.0 Silicon R47V50 16 12 29 22 20.8 14 1.0 Silicon R47V50 16 12 29 22 20.8 15 1.0 Silicon R47V50 16 12 29 22 20.8 16 6.4 Silicon Water 33 37 55 88 72.0 17 6.4 Silicon Water 33 37 55 88 72.0 18 6.4 Silicon Water 33 37 55 88 72.0 19 6.4 Silicon Water 33 37 55 88 72.0 20 1.6 Silicon Water 33 37 55 88 72.0 21 1.6 Silicon Water 33 37 55 88 72.0 22 1.6 Silicon Water 33 37 55 88 72.0 23 1.6 Silicon Water 33 37 55 88 72.0

V (µL) 0.1 0.2 0.5 0.1 0.2 0.5 0.1 0.2 0.5 0.1 0.2 0.5 0.1 0.2 0.5 0.1 0.2 0.5 1.0 0.1 0.2 0.5 1.0

18.2 Preliminary Results: Validation of the Simulation Code 1.8

167

x 10−3

1.6 1.4

Force [N]

1.2 1 0.8 0.6 0.4 0.2 0

0

2

4 6 8 10 12 Experiment number (R47V50)

14

16

Fig. 18.4. Comparisons between experiments, analytical approximation, and simulations (with silicone oil): closed circle, experimental points; plus, limits of the error intervals; open circle, analytical approximations with θA and θR ; open down triangle, simulation with θA ; open triangle, simulation with θR

5

x 10−3

Force [N]

4

3

2

1

0 16

17

18 19 20 21 22 Experiment number (Water)

23

24

Fig. 18.5. Comparisons between experiments, analytical approximation, and simulations (with water): close circle, experimental points; plus, limits of the error intervals; open circle, analytical approximations with θA and θR ; open down triangle, simulation with θA ; open triangle, simulation with θR

168

18 Results

where k is the stiﬀness of the blade, δe is the calibration deﬂection (i.e., the measured deﬂection of the beam when a known mass me is put on the beam), δ is the beam deﬂection due to the capillary force. The relative error on the mass me is supposed to be smaller than the other error sources and consequently set to zero. The typical error on the stiﬀness is due to the positioning error of the calibration mass (typically about 20 µm). As the calibration deﬂection is typically 600 µm, this leads to a relative error on the stiﬀness of about 3.3%. Finally, the deﬂection error is estimated at about 10 µm and is divided by the smaller measured deﬂection δi . For each simulation, the following information is plotted just right to the experimental points: •

• •

The analytical approximation is plotted for advancing and receding contact angles (◦) – the upper circle corresponds to the receding contact angle (larger force) The simulation result with the advancing contact angles ( ) The simulation result with the receding contact angles ( )

It can be seen on these pictures that the experimental results can be predicted by simulation for the experiments led with silicone oil, but in the case of water, the simulated force with a receding contact angle is much too large: the experimental value lies between the forces corresponding to the advancing and the receding contact angles in simulations [20–23] but is a little smaller in experiments [16–19]. The general trend is that the actual force rather corresponds to advancing contact angles (this is discussed again in Sect. 18.3). The inﬂuence of the volume (that cannot be predicted by the analytical approximation: the underlying cylindrical approximation of the meniscus makes the force independent of the volume of the liquid bridge) is diﬃcult to determine from our results: if there is an inﬂuence1 , it is smaller than our experimental and simulation accuracies. What can be observed is the inﬂuence of the gripper radius: the larger it is the larger is the force as indicated by the comparison between experiments [1–3] and [4–6], experiments [10–12] and [7–9; 13–15] and, ﬁnally, experiments [16–19] and [20–23]. The following sections are now devoted to the exploitation of both the simulation and the experimental tools.

18.3 Advancing vs Receding Contact Angle For liquid–solid combinations with small contact angles and low hysteresis, the capillary force does not depend too much on the angle as explained in the case of the sphere–plane interaction by the following equation: cos θR FR = . FA cos θA 1

Bhushan [24] cites Mazzone et al. [127]: “it can be shown that the force decreases with the volume”.

18.3 Advancing vs Receding Contact Angle

169

In the case of silicone oil on a silicon component, θA = 29◦ and θR = 16◦ , leading to FR /FA = 1.10. However, for solid–liquid combinations with larger angles and hysteresis such as, for example, water on silicon, θadv = 55◦ and θrec = 33◦ can lead to FR /FA = 1.46. The choice of the right angle must consequently be addressed carefully as illustrated by the following experiment, involving a spherical gripper GS-St-12.7 and a silicon component. The liquid is water and the used volumes [0.1, 0.2, 0.5, 1] µL (see experiments [16-19] of Fig. 18.5): the experimental results lead to F ≈ 1.34 mN ± 0.08. Let us now consider the contact angles (A states for “Advancing angle assumption” and R for “Receding angle assumption”): Table 18.2 indicates that the angle to be taken into consideration seems to be the advancing one. At ﬁrst sight the upward motion of the gripper during the picking step can be thought to lead to a receding motion of the liquid and consequently to a receding angle (Fig. 18.6a). However, come into close contact with the component, the approaching motion of the gripper pushes the liquid outwards, with an advancing contact angle (Fig. 18.6b). At the beginning of the upward motion of the gripper, the angle is still the advancing one because the gripper (and consequently the liquid too) has not yet begun to move (Fig. 18.6c). Only when the gap has begun to increase the contact angles move from advancing to receding ones (Fig. 18.6d). This assumption has been validated by direct observation of the scene with the CCD camera. θ1 and θ2 have been measured 15 times in picking situation, leading to the following (averaged) results: θ1 ≈ 57◦ ± 16 and θ2 ≈ 83◦ ± 6 (angles to be compared with the 55◦ and 88◦ of Table 18.2). Table 18.2. Measured contact angles of the steel–water and the silicon–water combinations Solid–Liquid combination Silicon–Water Steel–Water cos θ = 12 (cos θ1 + cos θ2 ) F = 4πγR cos θ

A θ1adv = 55◦ θ2adv = 88◦ 0.304 1.75 mN

R θ1rec = 33◦ θ1rec = 37◦ 0.819 4.71 mN

gap

(a)

(b)

Fig. 18.6. Advancing vs receding contact angles

(c)

(d)

170

18 Results

18.4 Inﬂuence of the Gap 18.4.1 Force–Distance Curve This section can be seen as an additional validation of the simulation or already be interpreted as a ﬁrst knowledge toward a gripper based in the surface tension eﬀects: the force–distance curve. This curve (Fig. 18.7) plots the capillary force exerted by a 7.9 mmΦ spherical gripper (GS-St-7.9) on a silicon component (Si-004-01). The force is exerted by a 0.5 µL silicone oil droplet (R47V50). This curve has been measured with an almost zero velocity (equilibrium curve). As far as the simulation is concerned, receding contact angles have been input since the meniscus is stretched by moving the gripper upwards. The correspondence between simulation and experiment can be seen on this picture, although the rupture distance predicted by the simulation is a little smaller than the measured one: this is discussed again in Sect. 18.11 devoted to the study of the separation distance. This result indicates that the simulation tool can predict the capillary force with separation distances different from zero: this comes as a complement to the previous validations made by comparing the simulation with the analytical approximations, which was valid only at contact. From the point of view of a capillary gripper design, this curves means that the sticking eﬀect due to the capillary force can be reduced or suppressed by increasing the distance between the gripper and the component. This separation could be achieved either by dynamical eﬀects (see also Sect. 18.10) or by pushing the component away from the gripper with

1

x 10−3 Simulation Experiments

Force [N]

0.8

0.6

0.4

0.2

0

0

0.2

0.4

0.6 Gap [m]

0.8

1 x 10−3

Fig. 18.7. Force–Distance curve for V = 0.5 µL R47V50, Si-component and GS-St7.9 (Both reprinted with permission from [108]. Copyright 2005 American Chemical Society.)

18.5 Inﬂuence of the Gripper Geometry

171

Ratio Force / Total

1

0.8

0.6

Laplace Tension Total

0.4

0.2

0

0

1

2

3 4 Gap [m]

5

6

7 x 10−4

Fig. 18.8. Respective contribution of the “tension” and “Laplace” terms in the total amount of the force—Ratios without dimensions (Both reprinted with permission from [108]. Copyright 2005 American Chemical Society.)

a less adhesive auxiliary tool or tip. The curve plotted in Fig. 18.7 helps to design the gripper by indicating the required range to separate the component from the gripper and/or the residual force. 18.4.2 Tension Force vs. Laplace Force The simulation results of Fig. 18.7 also allow to calculate and compare the importance of the interfacial tension force and the Laplace term of the capillary force (see Fig. 18.8). The results presented in this ﬁgure justify some approximations found in the literature, neglecting the “tension” term for small gaps. Nevertheless, this assumption is no longer valid for larger gaps and the tension term even becomes dominant.

18.5 Inﬂuence of the Gripper Geometry The inﬂuence of the gripper geometry involves the study of the inﬂuence of the radius in the case of a spherical gripper or the angular aperture p in the case of a conical gripper. The results of Sects. 18.2.2 and 18.2.3 indicate that the force is proportional to the gripper radius R as mentioned by the Israelachvili [89] approximation: F = 4πγR cos θ

(18.4)

172

18 Results FRef = 0.87mN

F0 [N]

10−3.1

Ref log10F0−log10Fref = 0.34 (log10V0 −log10 Vref)

10−3.3

−9

VRef = 10 m

10−3.5 10−11

10−10 V, Volume of liquid [m3]

3

10−9

Fig. 18.9. Inﬂuence of the angular aperture p for conical grippers: comparison between simulation (solid lines) and experiments led with 0.1 µL (open circle), 0.2 µL (open square), and 0.5 µL (open triangle) silicone oil (R47V50) with conical gripper with p = 5◦ (GC-St-5), p = 10◦ (GC-St-10), and p = 45◦ (GC-St-45) (Both reprinted with permission from [108]. Copyright 2005 American Chemical Society.)

In this case the volume of liquid plays no (or a minor) role. The situation is diﬀerent in the case of a conical gripper as indicated in Fig. 18.9, where it can be seen that the larger the volume of liquid, the larger the capillary force. This inﬂuence is nevertheless less signiﬁcant than that of the angular aperture: when the conical gripper diverges from the cylindrical tip (p = 0), the force dramatically decreases. This result, however, indicates that a geometry tip modiﬁcation could be used to successfully decrease the force and hence achieve the release task of a component caught by a capillary gripper. Figure 18.9 quantiﬁes the inﬂuence of this geometry change.

18.6 Inﬂuence of the Surface Tension The inﬂuence of surface tension has been investigated by simulation, for conical (GC-St-5) and spherical (Φ26 mm) grippers. For both grippers, the gap was equal to zero and the gripper contact angle was set to θ2 = 20◦ . To achieve this, it must be assumed that the gripper material changes when the liquid (i.e., surface tension) changes. The volume of liquid was set to V = 0.5 µL. Results are presented in Figs. 18.10 and 18.11. One can see the linear inﬂuence of the surface tension on the capillary force. This result can be explained by (6.11) FC = FT + FL = 2πr1 γ sin(θ1 + φ1 ) + 2γHπr12 .

(18.5)

18.6 Inﬂuence of the Surface Tension 6

x 10−3 θ1= 208

5 Force [N]

173

θ1= 308 θ1= 408 θ1= 508 θ1= 608

θ1= 108

4

θ1= 708

3 2 1 0 10

20

30

40 γ [mN−1]

50

60

70

Fig. 18.10. Inﬂuence of surface tension for diﬀerent contact angles θ1 – Conical tip (p = 5◦ ), Vol = 0.5 µL, Gap = 0 µm, θ2 = 20◦

0.012 θ1= 108

Force [N]

0.01

θ1= 208

θ1= 308 θ1= 408 θ1= 508 θ1= 608 θ1= 708

0.008 0.006 0.004 0.002 0 10

20

30

40 γ [mNm−1]

50

60

70

Fig. 18.11. Inﬂuence of surface tension for diﬀerent contact angles θ1 – Spherical tip (R = 13 mm), Vol = 0.5 µL, Gap = 0 µm, θ2 = 20◦

In this equation, θ1 is imposed by the couple component material/liquid and φ1 = 0 for ﬂat components. r1 and H are related to the meniscus geometry. Since this geometry must respect (8.13) −

1 pin − pout r , + = γ (1 + r2 )3/2 (1 + r 2)1/2

(18.6)

it means that for identical boundary conditions (i.e., identical contact angles, gap, and volume), diﬀerent surface tensions γ1 and γ2 will give the same geometry with two diﬀerent pressure diﬀerences ∆p1 and ∆p2 on the condition that ∆p1 ∆p2 = . γ1 γ2

(18.7)

174

18 Results x 10−5 7 6

z [m]

5 4 γ = 72e−3 Nm−1

3

γ = 10e−3 Nm−1

2 1 0 −1 1.18

1.2

1.22

1.24

1.26

1.28

r [m]

1.3 x 10−3

Fig. 18.12. Absence of inﬂuence of γ on the meniscus shape Table 18.3. Comparison between two menisci got with two diﬀerent surface tensions ∆p (Pa) Rneck (mm) ∆p γ

γ = 72 × 10−3 N m−1 −2451.51 1.236 34.049

γ = 10 × 10−3 N m−1 −340.327 1.235 34.033

This is illustrated in Fig. 18.12 in the case of a cylindrical gripper with a gap z = 50 µm and contact angles θ1 = θ2 = 30◦ . The comparison between these two menisci (Table 18.3) illustrates the fact that the same geometries lead to the same r1 and H, and consequently, the force is proportional to the surface tension.

18.7 Inﬂuence of the Contact Angle θ1 The inﬂuence of the contact angle θ1 has been studied by simulation for a conical (GC-St-5, p = 5◦ ) and spherical (R = 13 mm) gripper with z = 0 and V = 0.5 µL. In these simulations, θ2 = 20◦ . The decrease of the force as long as θ1 increases can be seen in Figs. 18.13 and 18.14. This behavior can be approximated by a cosine function in the case of the spherical gripper (Fig. 18.14).

18.8 Inﬂuence of the Relative Orientation The inﬂuence of the relative orientation of the gripper with respect to the component could not be simulated due to the axially symmetry assumption. Consequently, this inﬂuence has been studied experimentally by tilting the

18.8 Inﬂuence of the Relative Orientation 6

x 10−3

5 Force [N]

175

γ = 70mNm−1

γ = 60mNm−1

4 3 2

γ = 50mNm−1 γ = 40mNm−1 γ = 30mNm−1 −1

γ = 20mNm

1 0 10

γ = 10mNm−1

20

30

40 θ1 [8]

50

60

70

Fig. 18.13. Inﬂuence of θ1 for diﬀerent surface tensions γ—Conical tip (p = 5◦ ), Vol = 0.5 µL, Gap = 0 µm, θ2 = 20◦ 0.012 0.01

Force [N]

0.008 γ = 70mNm−1 γ = 60mNm−1

0.006

γ = 50mNm−1 γ = 40mNm−1

0.004

γ = 30mNm−1 γ = 20mNm−1

0.002

γ = 10mNm−1

0 10

20

30

40 θ1 [8]

50

60

70

Fig. 18.14. Inﬂuence of θ1 for diﬀerent surface tensions γ – Spherical tip (R = 13 mm), Vol = 0.5 µL, Gap = 0 µm, θ2 = 20◦

gripper with respect to the component, thanks to a manual rotational stage with 0.04◦ accuracy. The results are presented in Fig. 18.15 for a steel cylindrical gripper, a steel component, and a 0.1 µL R47V50 droplet. It can be seen that the relative orientation plays a major role on the force: this can dramatically inﬂuence the performances of a capillary gripper due to some unavoidable machining error or misalignment. On the contrary, this eﬀect can be used to force the release of the component after it has been positioned. Therefore, the gripper should be designed to allow a rotation degree of freedom. A rotation range of 10◦ can already decrease the force by a factor 5. Finally, let us emphasize the fact that these results are quite similar to those

176

18 Results 7

x 10−3

6

Force [N]

5 4 3 2 1 0

0

5

10

15 20 Tilt angle [8]

25

30

35

Fig. 18.15. Inﬂuence of the relative orientation of a steel cylindrical gripper (except for a tilt angle of 2◦ , the experimental points are within the interval error) Q N

F

A

F

Q

N

B W

C

α W

α

W

α

(a) Fig. 18.16. Tilt release

presented for van der Waals forces in [107]. To experiment this strategy, the component was picked (Fig. 18.16a) and moved to its ﬁnal location. Then the substrate was tilted with an angle α ≈ 6◦ and the component pushed against the substrate (Q in Fig. 18.16b): initially, Q is applied in A but due to the moment of the force N , the component begins to rotate around C and Q moves from A to B. Therefore, the meniscus moves to B too and the capillary force F is reduced because the gripper is no longer parallel to the component. If F becomes smaller than the weight W of the component, the releasing operation can be proceeded. The tilt angle α has to be chosen according to the component weight and to the desired force reduction (Fig. 18.15).

18.9 Auxiliary PTFE Tip Among several release strategies, the use of an auxiliary thin tip has already been envisaged [191]. Quantitative assessment of this strategy has been led with a PTFE conical tip with an angular aperture p = 70◦ . The results are

18.10 Dynamical Release

177

x 10−4 1.2

Force [N]

1 0.8 0.6 0.4 0.2 0

0

1

2

3

4

5

6

7

Experiment number ◦

Fig. 18.17. Force of a p = 70 PTFE conical tip on a silicon component (the experimental points are within the interval error)

presented in Fig. 18.172 . The ﬁrst three tests have been led with silicone oil on a silicon component (V = [0.1, 0.2, 0.5] µL) and the last four ones with water on a silicon component (V = [0.1, 0.2, 0.5, 1] µL). The results show that the forces are between 42 and 65 µN for silicone oil and between 50 and 90 µN for water, i.e., of at least two orders of magnitude lower than the forces exerted by the spherical grippers. Consequently, the substitution of materials and shape could be very successful to achieve the release task.

18.10 Dynamical Release 18.10.1 Simulation Results As already mentioned in [149], the release task in handling with adhesion force can be achieved dynamically, by giving the gripper an upward acceleration a large enough to detach the component. The previously described static simulation has been used to compute the capillary force exerted on 2

The error intervals are diﬀerent because two diﬀerent blades have been used to avoid mixing water and silicone oil. Therefore, the cantilever length was diﬀerent, leading to diﬀerent stiﬀnesses: 0.755 N m−1 for experiments [1–3] and 3.561 N m−1 for experiments [40–7]. Since a larger stiﬀness induces lower deﬂections, the relative errors as far as deﬂections are concerned are larger in experiments [4–7].

178

18 Results

(a)

(b)

(c)

(d)

Fig. 18.18. Manipulation by adhesion: (a) approach; (b) picking; (c) depose; (d) release

the component at each time step and hence deduce its kinematics. The canonical task is described in Fig. 18.18: the gripping phase is based on the adhesion force (b) and the release (d) is achieved by giving to the gripper an acceleration. Several simulations have been led with a conical gripper (p = 0.1), equal contact angles θ1 = θ2 = 30◦ and 0.28 µL water. The component was given a mass between 25 and 200 mg: consequently, we will see an evolution from perfect picking for the lighter mass3 to an impossible picking (i.e., a perfect release) for the heaviest mass. To do this, the kinematics of the gripper (freely determined by the user) was supposed to be a trapezoidal distribution of velocity: ﬁrst an increase ramp limited by a user deﬁned acceleration a until the velocity reaches the prescribed value. Then the velocity is kept constant until the gripper is decelerated (deceleration = −a) to reach a zero velocity at the end of the predeﬁned time interval. An example is given in Fig. 18.19 where it can be seen that the time interval has been set to 10 ms, the acceleration to a = 100 m s−2 , and the velocity to v = 0.1 m s−1 (the position of the gripper is deduced from this information by assuming an initial position equal to zero: we see a parabolic trend between t = 0 ms and t = 1 ms, corresponding to a linear velocity, and a linear evolution corresponding to the constant value of the velocity between t = 1 ms and t = 9 ms. Between t = 9 ms and t = 10 ms, the position z of the gripper shows a parabolic behavior again). The simulation has been launched for three diﬀerent values of the component mass: 25, 50, and 200 mg. Each time, the kinematics of the gripper (deﬁned by the user) has been superposed to that of the component (computed), the gap between the component and the gripper, and the corresponding value of the capillary force have been plotted. Finally, the normal reaction N of the substrate below the component is also represented (=contact force). The case of a perfect picking is illustrated for m = 25 mg in Fig. 18.19, where the position, velocity, and acceleration of the component are equal to those of the gripper. This is conﬁrmed in Fig. 18.20 by a gap and a contact force N equal to zero: this means that the component sticks to the gripper all the time; the value of the 3

Here the term “light” must be related to the acceleration a. For lower acceleration, a heavier mass will also be considered as light enough.

18.10 Dynamical Release

z [mm]

1

179

Gripper Object

0.5 0

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5 6 Time [ms]

7

8

9

10

vz [ms−1]

0.1 0.05 0

vz [ms−2]

100 0 −100

Fig. 18.19. m = 25 mg, V = 0.28 µL water, θ1 = θ2 = 30◦ , conical gripper (p = 0.1): positions, velocities, and accelerations

Gap[mm]

1

0

Contact force [mN]

Capillary Force [mN]

−1

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

5 4 3 2 0.4

0.2

0

Time [ms]

Fig. 18.20. m = 25 mg, V = 0.28 µL water, θ1 = θ2 = 30◦ , conical gripper (p = 0.1): gap, capillary, and contact forces

180

18 Results

z [mm]

1 Gripper Object

0.5

0

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

vz [ms−1]

0.1

0 − 0.1

az [ms−2]

100 0

−100

Time [ms]

Fig. 18.21. m = 200 mg, V = 0.28 µL water, θ1 = θ2 = 30◦ , conical gripper (p = 0.1): positions, velocities, and accelerations

capillary force F = 3.70 mN is well larger than the weight of the component W = 0.25 mN. The opposite case is the perfect release presented in Fig. 18.21 for m = 200 mg: the component begins to move upwards (at time t = 2.5 ms, its velocity is equal to zero and its position is 10 µm above the substrate) before falling and reaching the substrate at time t = 4.2 ms (this time is conﬁrmed by the apparition of a contact force N at t = 4.2 ms. Note that N is equal to the weight W = 1.96 mN of the component). The small upward motion is unavoidable since the component is initially in contact with the gripper, leading to an initial force F = 3.7 mN larger than the weight. The only way to avoid this is to have a capillary force at contact smaller than the weight, but in this case, the picking task cannot be performed. However, the gap increases very quickly and the corresponding capillary force becomes smaller than the weight at time t = 1.1 ms: this can be seen by observing that at this time the force F is equal to the component weight W (Fig. 18.22) or by observing that the acceleration of the component becomes equal to zero at this time (Fig. 18.21). A last remark on Fig. 18.22 is the switch in capillary force at time t = 3.4 ms from F = 0.41 mN to zero: this change is forced once the capillary force becomes smaller than a predeﬁned threshold (typically 10% of the force computed at contact). Between these two extreme situations, an evolution can be observed from the perfect pick to the perfect release in Fig. 18.23. The latter ﬁgure is particularly interesting because the component begins to be released: indeed

18.10 Dynamical Release

181

Gap[mm]

1

0.5

Contact force [mN]

Capillary Force [mN]

0 0

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

6

7

8

9

10

4

2

0 0 2

1

0 0

Time [ms]

Fig. 18.22. m = 200 mg, V = 0.28 µL water, θ1 = θ2 = 30◦ , conical gripper (p = 0.1): gap, capillary, and contact forces

z [mm]

1 Gripper Object

0.5

0

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

vz [ms−1]

0.2

0.1

0

az [ms−2]

100

0 −100

Time [ms]

Fig. 18.23. m = 50 mg, V = 0.28 µL water, θ1 = θ2 = 30◦ , conical gripper (p = 0.1): positions, velocities, and accelerations

182

18 Results

Gap[mm]

0.04

0.02

Contact force [mN]

Capillary Force [mN]

0

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

4

3

2 1

0.5

0

Time [ms]

Fig. 18.24. m = 50 mg, V = 0.28 µL water, θ1 = θ2 = 30◦ , conical gripper (p = 0.1): gap, capillary, and contact forces

Fig. 18.23 indicates that until time t = 1 ms its acceleration is positive (the component moves upwards and leaves the substrate as indicated by N = 0 in Fig. 18.24) but smaller than that of the gripper. It can also be seen that its velocity is smaller than that of the gripper until t = 2 ms. Nevertheless, in time t = 1 ms, the acceleration of the gripper is set to zero, so that between t = 1 ms and t = 2 ms, the velocity of the component can reach up to that of the gripper: the gap is consequently maximum at t = 2 ms as indicated in 18.24. Between t = 2 ms and t = 3.5 ms, the velocity of the component is larger than that of the gripper: the gap decreases and the capillary force increases. After t = 3.5 ms, the component perfectly follows the motion of the gripper. This ﬁgure indicates that the component would be ﬁnally released if the acceleration duration of the gripper was larger, but at a height z at least higher than 30 µm (this is the position of the component at time t = 1 ms). 18.10.2 Experimental Results For the experimental testing of the dynamic release, a speciﬁc workbench has been set up in collaboration with Maxime Frennet [61]. As shown in Sect. 18.9, if the gripper acceleration is stopped too early, a component that has begun to detach can stick to the gripper again due to its acquired upward velocity (Fig. 18.24). Consequently, the actuator has been programed in order to ﬁrst displace downwards to give the component a downwards velocity before

18.10 Dynamical Release

183

ε

F

ε

W

(a)

F

W

(b)

(c)

Fig. 18.25. Sources of errors in the tests for dynamic release. (a) Misalignment ; (b) Droplet positioning error; (c) Axial symmetry error

beginning its upward acceleration : the full acceleration lies in the range between 108 and 123 m s−2 and is applied during 10 ms. Then the control mode of the actuator must be switched from the “acceleration” mode to the “position” mode and the actuator decelerates before stabilizing about the position at the switch time. Several gripper tips have been tried on this actuator (conical and spherical grippers in steel, foam plane gripper); the tests led with the conical and spherical grippers did not give satisfaction, not because they could not show the ability to catch and release components but well because the actual forces seemed to be lower than the predicted ones. This could be explained by misalignments or axial asymmetry as shown in Fig. 18.25: they usually induce the rotation of the component along the gripper, leading to unexpected conﬁgurations. The components used for these tests were the components St-i and Si-i deﬁned in Sect. 17.9.2. To increase the repeatability of the handling task, a foam gripper has been tried: it consisted of a small foam parallelepiped (2 × 2.5 × 2 mm3 , Length–Width–Thickness) glued at the end of a support screwed in the movable axis of the actuator. As this kind of gripper has not been modeled, the force has been measured separately on a silicon component (Si-004-1) with 1 µL R47V50. The gripper has been dipped once to a 1 µL silicone oil droplet. Then, after each measure, the component was dried and the gripper applied again on the component with a force of about 0.86 mN before measuring the capillary force linking the foam gripper and the silicon component. This experiment has been led 30 times (results are presented in Fig. 18.26). A second experiment consisted in measuring the force seven times by dispensing a new 1 µL droplet before each new trial. The mean value of these seven trials (F = 0.277 mN) has been used to roughly predict if a component would stick or not to the gripper tip during a trial of dynamic release. A range has consequently been deﬁned for the masses: F a+g F = . g

mmin =

(18.8)

mmax

(18.9)

184

18 Results 4

x 10−4

3.5 Available Force [N]

3 2.5 2 1.5 1 0.5 0

0

5

10 15 20 Number of pick operations

25

30

Fig. 18.26. Capabilities of a foam gripper 1.5 Lower Boundary of the Working Domain, m=2.5mg Upper Boundary of the Working Domain, m=28.2mg

1 Shift between prediction and experiments

Picked but not released Picked and Released Not Picked

0.5

0 0

50

100 Mass [mg]

150

200

Fig. 18.27. Results of the dynamical handling

If a component has a mass smaller (larger) than mmin (mmax ), it cannot be released (picked up). With a = 100 m s−2 , this led to mmin = 2.5 mg and mmax = 28.2 mg (represented by the solid lines in Fig. 18.27). Then, experimental tests have been led with diﬀerent masses: 12.8, 15.1, 17.1, 46.1, 54.1, and 183.1 mg: • • •

Components with m = 12.8 and 15.1 mg could not be released Components with m = 17.1 and 46.1 mg could be picked up and released Components with m = 54.1 and 183.1 mg could be picked up

18.11 Approaching Contact and Rupture Distances

185

The predicted results are not perfectly correct (there is a shift of about 23 mg with the experimental results), but the trend is indeed obtained experimentally: it can be distinguished between three intervals, indicating that a handling window exists either by adapting the mass for a given gripper acceleration or by adapting the acceleration for a given component. Nevertheless, this release method leads to a poor positioning accuracy. Indeed, in successful release trials, the component could touch the substrate at a diﬀerent position from that where it was deposed by the gripper. This positioning error can reach several tenths of millimeter, which is unavoidable. The situation was hardly improved by increasing the downwards impulsion. The conclusion of these tests are the following: • • •

There is actually a handling window in dynamical manipulation The positioning accuracy is not good (several tenths of millimeter), at least for millimetric components Consequently, this method cannot be used without a dramatical improvement of the accuracy (in particular, the alignment must be achieved very carefully (Fig. 18.25). Maybe at lower scale, higher accelerations can be reached with piezoelectric actuators, but it should be checked if the relative positioning error (i.e., error divided by component size) is actually decreased

18.11 Approaching Contact and Rupture Distances To investigate the approaching and rupture distances, the following protocol (Fig. 18.28) has been applied to a spherical steel gripper with radius R = 1.6 mm and a silicon component, with volumes V = [0.1, 0.2, 0.5, 1] µL of water and silicone oil: 1. The amount of liquid is dispensed directly on the component (Fig. 18.28a) 2. The gripper is moved downwards until contact with the droplet is detected: this gives the initial approach distance (Fig. 18.28b)

Rupture distance(1)

Initial approach distance

(a)

(b)

(c)

Rupture distance(2)

(d)

Fig. 18.28. Rupture protocol: (a) liquid dispensing; (b) initial contact; (c) rupture; (d) second approach

18 Results 1200

1200

1000

1000

Distance [µm]

Distance [µm]

186

800 600 400 200 0

800 600 400 200

0

0.2 0.4 0.6 0.8 Volume of water [µL]

(a)

1

0

0

0.2

0.4

0.6

0.8

1

Volume of R47V50 [µL]

(b)

Fig. 18.29. Approaching and rupture distances. Experiments: initial approach (plus), second approach (open triangle), ﬁrst rupture (open square), second rupture (asterisk ). Simulations: initial approach (dotted line), second approach (dot dashed line), rupture (dashed line). As a comparison, the solid line states for z = V 1/3 . (a) Water; (b) R47V50

3. The liquid bridge is stretched until rupture: this is called the rupture distance 1 (Fig. 18.28c) 4. After the ﬁrst rupture, the volume of liquid is now distributed over both the gripper and the component: the approaching distance has now changed into the approaching distance 2 (Fig. 18.28d) 5. The gripper is moved upwards and the rupture gap is measured for the second time The results presented in Fig. 18.29 show that the rupture gap is slightly smaller at the second bridge collapse. The diﬀerence is between 6.5% and 9.9% for water, 2.0% and 5.7% for silicone oil. Simulations give the right trend and the order of magnitude although they are not so accurate. The reason therefore should be further investigated. Nevertheless, the simulation provides an order of magnitude for gripper design: the approach and rupture distances supply information on the required displacement range of the gripper for picking (approaching distance) and release (rupture distance). If the gripper could be “moved away” from the component, release would occur by meniscus collapse. This could be achieved by using an auxiliary tip with hydro (oleo)phobic properties.

18.12 Shear Force The shear force exerted by the meniscus has been tested experimentally (see Fig. 18.30) because this test is out of the scope of the axial symmetry assumption. Trials have been led with water, a silicon component, and a steel spherical gripper R = 6.35 mm. The gap was about z = 200 µm. In comparison, the

18.13 Conclusions

187

x 10−4

Shear force [N]

2.5 2 1.5 1 0.5 0

0

1

2 3 4 Volume of water [m3]

5 x 10−10

Fig. 18.30. Shear force (experimental points and their averages)

forces generated in the z direction in the same conditions are, respectively, F = 0.35 × 10−3 N, F = 0.58 × 10−3 N, and F = 1.00 × 10−3 N. The norm of the shear force to overcome is consequently reduced to 28.7%, 24.1%, and 20.0% of the force along z. If the gripper is moved perpendicularly to the z direction, the contact force must also be taken into account. In some cases, the viscous drag is not neglected [25]. Here the measure only concerns the shear capillary force because there was no contact between the gripper and the component (gap z = 200 µm) and the shear velocity was almost zero.

18.13 Conclusions This chapter has shown that the models proposed in Part II were in good agreement with experiments and analytical benchmarks when available. The results presented in this chapter have illustrated the inﬂuence of each parameter on the capillary force. These results will be summarized in Chap. 21, together with the results of the case study. It will allow to sketch some design rules for a surface tension gripper.

19 Example 3: Application to the Watch Bearing Case Study: Characterization

19.1 Introduction This chapter describes the characterization of the required parameters to be used in the force model of Chap. 10. It focuses on the available grippers, the diﬀerent components, the kinds of liquids used in the micromanipulation and their properties (surface tension, viscosity, and density). The liquid dispensing method is also addressed and, ﬁnally, the measurement of the contact angles and the contact angle hysteresis are described. Based on a Anova analysis, the importance of the coating on the repeatability of the contact angles is shown (the inﬂuence is here clearly negative to an unadapted choice of the coating).

19.2 Available Grippers According to the speciﬁcations given in Chap. 10, twelve grippers have been manufactured in stainless steel (sketches are given in Fig. 19.2). These grippers vary from one another concerning the presence/absence of an internal channel initially foreseen for liquid supply (it will be shown later that it is unnecessary), the diameter (0.3 mm or 0.5 mm), the presence or absence of a hydrophobic coating, and the shape of the extremal tip (with or without the conical concavity which is preferable to ensure centering (see Fig. 19.1)). A summary of their properties is given in Table 19.1. The geometrical characteristics of these 12 grippers have been measured using the control points deﬁned in Fig. 19.2. The measured values for the 0.5 mm (0.3 mm) diameter grippers are presented in Table 19.2 (19.3).

190

19 Example 3: Application to the Watch Bearing Case Study

(a)

(b)

Fig. 19.1. Pictures of two grippers. (a) Gripper E with a cylindrical tip without concavity, an internal Φ0.1 mm channel and no coating (the two marks below the gripper are separated from 1 mm); (b) Gripper B with a concavity, an internal Φ0.1 mm channel and an hydrophobic coating Table 19.1. Properties of the grippers A B C D E F G H I J K L

Diameter (mm) 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.3 0.3 0.3 0.3

Channel Yes Yes No No Yes Yes No No No No No No

Concavity No Yes No Yes No Yes No Yes No Yes No Yes

Hydrophobic coating Yes Yes Yes Yes No No No No Yes Yes No No

αg

αd y

t w z

x

Fig. 19.2. Measured points

19.4 Liquid Properties

191

Table 19.2. Characterization of the 500 µm diameter grippers A B C D E F G H Average Std

w (µm) tmin (µm) tmax (µm) x(µm) y(µm) z(µm) 494.8 907.1 917.0 448.6 511.3 864.3 537.7 864.3 870.9 537.7 527.8 834.6 491.5 1091.9 1101.8 494.8 498.1 1082.0 491.5 950.0 956.6 484.9 498.1 923.6 504.7 851.1 834.6 494.8 501.4 646.5 527.8 903.8 907.1 508.0 481.6 897.2 494.8 1095.2 1105.1 491.5 494.8 1098.5 511.3 966.5 983.0 501.4 501.4 963.2 506.8 953.7 959.5 495.2 501.8 913.7 17.6 94.5 100.0 24.8 13.4 144.1

αl (◦ ) αr (◦ ) 135.5 136.0 134.5 135.4 140.0 139.7 138.9 140.5 134.9 138.9 135.0 135.6 137.7 138.9 139.4 140.1 137.0 13801 2.3 2.1

Table 19.3. Characterization of the 300 µm diameter grippers w (µm) tmin (µm) tmax (µm) x(µm) y(µm) I 1052.3 1078.7 300.2 296.9 J 316.7 1062.2 1065.5 316.7 303.5 K 313.4 1042.4 1055.6 303.5 293.6 L 283.7 1016.0 1025.9 296.9 287.0 Average 304.6 1043.2 1056.4 304.3 295.2 Std 18.2 19.9 22.4 8.7 6.9

z(µm) 992.9 1016.0 1022.6 963.2 998.7 26.80

αl (◦ ) 136.8 139.6 140.1 139.1 138.9 1.5

αr (◦ ) 142.8 140.4 141.5 140.5 141.3 1.1

Table 19.4. Surface tension Liquid Isopropanol Water L23

Surface tension (N m−1 ) 23.2 × 10−3 73.0 × 10−3 33.5 × 10−3

19.3 Available Components The components handled in this application are the Zirconium balls of the chosen ball bearing. Two kinds of diameters are available, 0.3 mm and 0.5 mm, to be picked by grippers with a corresponding diameter.

19.4 Liquid Properties The properties that play a role in the presented models are the surface tension γ (capillary force models) on the one hand and the dynamic viscosity µ and the density ρ (characteristic times) on the other hand. The surface tensions have been measured by the Du Nou¨ y ring method [1] applied, thanks to a Sigma 703 sensor: they are presented in Table 19.4. The densities have been measured from the volume and the mass. Results are given in Table 19.5.

192

19 Example 3: Application to the Watch Bearing Case Study

Table 19.5. Density Liquid Isopropanol Water L23

Density (kg m−3 ) 768 1000 906

Source Own experiment Classical value Own experiment

Table 19.6. Dynamic viscosity Liquid Isopropanol Water L23

Dynamic viscosity at 20◦ C(Pa s) 2.27 × 10−3 1 × 10−3 21.7 × 10−3

Source [62] Classical value MPS (24 cs) and density

Finally, the values of the dynamic viscosity, given by the suppliers, are given in Table 19.6.

19.5 Liquid Dispensing It has ﬁrst been envisaged to supply the liquid through the internal channel, but it turned out that this method would lead to a complicated pressure control solution, an instability of the hanging droplet and a conﬁguration much more diﬃcult to model (in the force model presented in Chap. 10, the pressure inside the meniscus is given by the external pressure and the pressure drop across the interface, which is given by the surface tension and the curvature. If the internal pressure is imposed, the volume of liquid has to be determined in order to get a meniscus whose shape ﬁts the contact angle as well as this – now imposed – pressure diﬀerence). The second solution would have been to extend the manual dispensing solution applied in the ﬁrst experiments of Chap. 18, but the main problem with water and isopropanol is the evaporation with the time. Let us note that it is not really a problem from an application point of view, because it is interesting to eliminate the residual traces of liquid on the component. It is, however, a drawback for gripper characterization because the evaporation process can be faster than the typical experiment duration. For example, we have experimented the following evaporation times for liquid droplets posed on a ﬂat substrate in the ambient conditions (see Fig. 19.3, 19.4 and 19.5): 1. 1 µL isopropanol evaporates in 15 s 2. 1 µL water evaporates in 10 min 3. Oil like R47V50 silicone oil or L23 synthetic oil does not evaporate or less than 1% Finally, the principle of liquid feeding which has been retained is to dip the gripper tip to a liquid tank. A small liquid droplet hangs at the bottom of the gripper, as illustrated in Fig. 19.6. One can expect the droplet height

19.5 Liquid Dispensing

(a)

(b)

(c)

(d)

193

(e)

Fig. 19.3. Evaporation of a water droplet (1). (a) t = 0 s; (b) t = 60 s ; (c) t = 120 s; (d) t = 190 s; (e) t = 250 s

(a)

(b)

(c)

(d)

(e)

Fig. 19.4. Evaporation of a water droplet (2). (a) t = 310 s; (b) t = 370 s; (c) t = 0430 s; (d) t = 500 s; (e) t = 560 s 20

V [nL]

15

10

5

0

0

100

200

300 400 Time [s]

500

600

700

Fig. 19.5. Water evaporation d

h

Fig. 19.6. Tip dipping d is the gripper diameter and h is the height of the hanging droplet

194

19 Example 3: Application to the Watch Bearing Case Study

Table 19.7. Anova of the hanging height model ¯ Average H A1 A2 Residual Total Data

SS 35.29 0.02 0.02 0.13 35.46

DF 1 1 1 356 359

MS 35.29 0.022 00.07 0.00037 -

F 95000 59 44 1 -

p < 10−9 < 10−9 < 10−9 -

h to depend on the following parameters: the liquid surface tension γ, the gravity g, the liquid density ρ, the gripper diameter D, the contact angle θ. Nondimensional formulated, it means that the ratio height/diameter can be expressed as follows: ρgd2 h = f (θ, ) = f (θ, Bo ). d γ

(19.1)

Therefore, the problem is much easier to represent graphically: there are two nondimensional input parameters and one nondimensional output parameter H = hd . Using the design of experiment theory, the linear model without interaction has been tested: ¯ + A1 θ + A2 Bo + , H=H

(19.2)

¯ A1 , and A2 have been calculated by least squares, where the coeﬃcient H, and the corresponding Anova is shown in Table 19.7. Using the genuine replications of the experiments (each point of the experimental space has been replicated 10 times), the residual sum of squares (0.13) has been split into the pure error (0.05) and the lack of ﬁt of the model (0.08). It is also possible to calculate a reliability interval for the coeﬃcient of the model as follows: 1. An estimation of the variance of the model is calculated: sˆ =

(y − yˆ) (y − yˆ) n−p

(19.3)

2. The impact of this estimated variance is ampliﬁed by the dispersion matrix: Vb = (X X)−1 sˆ

(19.4)

3. Finally, SEb =

diag(Vb )

(19.5)

19.6 Contact Angles

195

These calculations lead to the following reliability intervals given in Table 19.8. ¯ The conclusion to be drawn is that except the main eﬀect given by H, all the other parameters play no role in the volume of liquid obtained by tip dipping. In other words, whatever the material and the liquid, the hanging droplet has a height determined by the gripper diameter.

19.6 Contact Angles To measure the contact angle hysteresis, the gripper was moved downwards (Fig. 19.7a) to measure the advancing contact angle and upwards (Fig. 19.7b) to measure the receding contact angle. Both angles were measured by moving the gripper very slowly (a few microns per second, in a quasi static conﬁguration). All combinations of liquids and of grippers have been considered [112], each angle being measured 10 times. The average and the standard deviations of each conﬁguration are plotted in Figs. 19.8a and b and 19.9a and b. In these ﬁgures, grippers A, B, C, D, I, J are coated and grippers E, F, G, H, K, L are not. From the detailed Anova analysis led in [112], it can be concluded that the inﬂuence of the applied coating is not signiﬁcant. Indeed, the main eﬀect Table 19.8. Error intervals ¯ Average H A1 A2

Value 0.31501 0.00191 −0.01298

Std 0.00126 0.00227 0.00195

θrec θadv

(a)

(b)

Fig. 19.7. Illustration of the contact angle hysteresis (case of a stainless steel Φ0.5 mm gripper and water) (Reprinted with permission from [112]. Copyright 2006 Institute of Physics.)

196

19 Example 3: Application to the Watch Bearing Case Study

100

Isopropanol L23 Water

80 60 40 20 0

Std advancing angles

Mean advancing angles

100 Isopropanol L23 Water

80 60 40 20 0

1 2 3 4 5 6 7 8 9 10 11 12 Grippers

A B C D I

(a)

J E F G H K L Grippers

(b)

Fig. 19.8. Advancing contact angles at latest 4 days after coating. (a) Average values; (b) standard deviations of each ten of trials – Coated grippers: A, B, C, D, I, J; noncoated grippers: E, F, G, H, K, L (Reprinted with permission from [112]. Copyright 2006 Institute of Physics.)

100 Isopropanol L23 Water

80 60 40 20 0

A B C D I J E F G H K L Grippers

(a)

Std receding angles

Mean receding angles

100

Isopropanol L23 Water

80 60 40 20 0

A B C D I J E F G H K L Grippers

(b)

Fig. 19.9. Receding contact angles at latest 4 days after coating. (a) Average values; (b) standard deviations of each ten of trials – Coated grippers: A, B, C, D, I, J; non coated grippers: E, F, G, H, K, L (Reprinted with permission from [112]. Copyright 2006 Institute of Physics.)

of the coating is of the same order of magnitude – but a bit smaller – than the main eﬀect of the liquids on the advancing contact angle (4 days after coating). It becomes totally negligible in the case of the receding contact angle. This data analysis leads to the conclusion that the coating of the grippers is not useful in our application since the main eﬀect of the coating is clearly negligible in comparison with the main eﬀect of the liquid. More physically, this coating has been optimized to be vaporized on a silicon substrate. This coating is just a monolayer, which can be damaged or pulled oﬀ at some locations, leading to a quite heterogeneous surface. This physical

19.6 Contact Angles

197

suggestion is reinforced by larger standard deviations in the presence of coatings. The conclusion of this analysis is that in case of applying any coating, the wear resistance should be addressed carefully since the repeated contact between the gripper and the consecutive components induce damage onto this gripper coating, leading to a lack of eﬃciency and an increase of the dispersion.

20 Example 4: Application to the Watch Bearing Case Study: Results

20.1 Introduction In Chap. 10, we have brieﬂy described the design of a surface tension based gripper to be used in the placement of 0.3- or 0.5-mm diameter balls of a watch ball bearing. Chapter 19 has described the characterization of this gripper and the contact angles made by liquids that could be used (isopropanaol, water, and oil). This chapter aims at detailing the results of the typical micromanipulation task, illustrated in Figs. 20.1 and 20.2. The 0.5-mm diameter ball “ﬂoats” by surface tension on a spherical cap-shaped water drop posed on a steel substrate (Fig. 20.1a), the gripper (ripper D, see Table 19.1 for details) is aligned above the ball (Fig. 20.1b) and then moved downwards until contact with the ball. The result of the pick operation can be seen in Fig. 20.1c. Then the gripper is moved toward the ball bearing (Fig. 20.1d) and positioned above the cavity in which the ball is to be placed (Fig. 20.2a). The ball is positioned in its ﬁnal location (Fig. 20.2b) and the release takes place by moving the gripper radially outwards (Fig. 20.2c and d).

20.2 Picking 20.2.1 Introduction The picking task cannot be separated from the feeding solution. Indeed, small components must ﬁrst be separated and positioned before picking. Nevertheless, this task has to be done whatever the picking principle is, and some solutions already exist: feeding trays, tapes, etc. As it is not the primary point of this book, it has been considered that at worst the position of the balls could be determined by vision. First some typical picking errors are presented in Sect. 20.3, before proposing solutions (Sect. 20.2.3) and tracks for automated control (Sect. 20.2.4).

200

20 Example 4: Application to the Watch Bearing Case Study: Results

(a)

(b)

(c)

(d)

Fig. 20.1. Picking sequence: (a) the ball is initially placed on a small droplet (bottom right); (b) alignment of the gripper; (c) ball picking; (d) displacement of the gripper toward the ball bearing (Reprinted with permission from [112]. Copyright 2006 Institute of Physics.)

(a)

(b)

(c)

(d)

Fig. 20.2. (a) Positioning; (b) Placing of the ball the cage of the bearing; (c) and (d) Release of the component by radial outwards motion of the gripper (Reprinted with permission from [112]. Copyright 2006 Institute of Physics.)

(a)

(b)

(c)

(d)

Fig. 20.3. Picking errors: (a) centering error; (b-c) balls get stuck after water evaporation; (d) surface forces curiosity (Φ0.5 mm Zr O2 balls, conical grippers) (Reprinted with permission from [112]. Copyright 2006 Institute of Physics.)

20.2.2 Errors Figure 20.3 illustrates some typical picking errors. The ﬁrst error (not shown in the ﬁgure) is that when no ball is picked up by the capillary gripper. It can happen when the gripper is not aligned with respect to the ball. There is a small centering eﬀect due to the conical geometry of the gripper, but we experimented that this centering eﬀect was limited by the friction. Usually, by adding a bit of liquid and by positioning the gripper properly this could be corrected. Then second error type is a lack of centering, illustrated in Fig. 20.3a. In this case, the ball must be released and the picking operation must be tried again. To release the ball, it is suﬃcient to dip it to a liquid tank: since there

20.2 Picking

201

Fig. 20.4. Top view of the hexagonal network of the 10 balls: The gray ball is picked by the gripper (Reprinted with permission from [112]. Copyright 2006 Institute of Physics.)

is no more liquid–gas interface between the ball and the gripper, there is no more capillary force and the weight of the ball pulls it downwards. When working with water, other complications can occur, as illustrated in Figs. 20.3b and c, showing the case of several balls sticking with each other, probably due to the capillary forces caused by residual traces of liquid.1 In some cases, these sticking balls form a regular hexagonal shaped alignment, as shown in Fig. 20.4. Finally, Fig. 20.3d illustrates the sensitivity of the micromanipulation task to its environment. The ball seems to be levitated but the underlying reason is the presence on a small dust linking the ball and the gripper, whose eﬀect dominates the weight of the ball. The phenomenon was very stable (a few minutes) , henceforth discarding eﬀects such as electrostatic levitation. 20.2.3 Solutions Diﬀerent solutions can be applied to these problems. The ﬁrst category of solutions aims at guaranteeing the picking of only one ball at a time. This can be best performed by posing the balls at the interface of a liquid, such as depicted in Fig. 20.5. These pictures illustrate the ability to pick one ball at a time from an initial alignment of four balls (Fig. 20.5a). The coated concave conical gripper is positioned above the second ball from left (Fig. 20.5b), moved downwards in contact with this ball (Fig. 20.5c). Figure 20.5d shows the pick operation of this ball. The pick operation is then improved with what can be called a kind of “liquid feeding.” The eﬃciency of this strategy clearly depends on the volume of liquid (which, for water, means dependent on time because of the evaporation), on the kind of liquid, and on the number of balls in the neighborhood of the picked ball. Indeed, in the case of regular balls network with the L23 oil, the picking eﬃciency was not clearly improved by 1

It has been shown the large evaporation rate in Fig. 19.3 but when the meniscus is trapped between two solid surfaces close to one another, it turns out a large reduction of the evaporation.

202

20 Example 4: Application to the Watch Bearing Case Study: Results

(a)

(b)

(c)

(d)

Fig. 20.5. Coated conical tip (B): picking of sorted balls from a liquid (Reprinted with permission from [112]. Copyright 2006 Institute of Physics.)

(a)

(b)

(c)

(d)

Fig. 20.6. Coated conical tip (B): picking of non sorted balls from a liquid (Reprinted with permission from [112]. Copyright 2006 Institute of Physics.)

the presence of a liquid ﬁlm. The reason for this probably lies in the fact that the contact angle between the ball (in zirconium) and the oil is much smaller than the one got with water. Therefore, the capillary force between balls can be stronger, the sum of the forces between the balls being larger than the force between the gripper and the picked ball. The pictures shown in Fig. 20.6 illustrate the “pick from wet bulk” strategy: a series of 10 balls ﬂoat by surface tension on a spherical cap-shaped water droplet put on a steel substrate. In Fig. 20.6, the gripper is dipped to the balls, and picks up only one ball (Figs. 20.6b and 20.6). The second category of solutions consists in releasing all the picked balls, either when there is a centering error or when more than one ball has been picked. A way to release all the balls is to dip the gripper to a liquid tank: since there is no capillary force in the liquid, the weight of the balls pull them downwards. 20.2.4 Automated Control The picking task can be controlled by vision, using an image analysis based on the recognition of the gripper blob (i.e., contour). The analysis routine starts from the black and white images of the grippers, such as shown in Fig. 20.7. The blob analysis is lead from the cropped area located just below the gripper tip, such as indicated in Fig. 20.8. Then, from the coordinates of the contour points of each blob, the area, perimeter, and compacity of the blob are calculated. Let us consider that a blob is deﬁned by its coordinates:

20.2 Picking

(a)

(b)

(c)

203

(d)

Fig. 20.7. Black and white pictures of four picking results. (a) Correct picking; (b) no ball is picked; (c) and (d) too many balls have been picked

(a)

(b)

(c)

(d)

Fig. 20.8. Corresponding blob analysis of the four picking results of Fig. 20.7

xi ,

1 ≤ i ≤ n,

(20.1)

yi ,

1 ≤ i ≤ n.

(20.2)

Then, the perimeter is given by P =

ds =

n−1

dsi =

i=1

n−1

(xi+1 − xi )2 + (yi+1 − yi )2 .

(20.3)

i=1

The area calculation is based on the Stokes theorem: ¯ ¯ × F¯ ) · dS¯ = F¯ · dS. (∇

(20.4)

Since the rotational of F¯ can be written in the cartesian coordinates as ∂Fz ¯ ∂Fx ¯ ∂Fx ∂Fy ¯ × F¯ = ( ∂Fy − ∂Fz )¯ 1x + ( − )1y + ( − )1z ∇ ∂z ∂y ∂z ∂x ∂x ∂y

(20.5)

and dS¯ = dS ¯ 1z ,

(20.6) ∂F

x if we choose Fx ≡ 0 and Fy ≡ x, we will have ∂xy − ∂F ∂y = 1 and ¯ ¯ ¯ (∇ × F ) · dS = dS = S = (Fx dx + Fy dy) = x dy.

(20.7)

Therefore, the area can be approximated by A=

x dy =

n−1 i=1

xi (yi+1 − yi ).

(20.8)

204

20 Example 4: Application to the Watch Bearing Case Study: Results 4

0.8

2

0.6

1.5

0.4

1

0.2

0.5

0

0

x 10

1000 800 600 400

1

2

3

4

(a)

200 1

2

3

0

4

(b)

1

2

3

4

(c)

Fig. 20.9. (a) Compacity (without dimension); (b) Area (pixel2 ); (c) Perimeter (pixel). Note that 1 = 1 ball, 2 = 0 ball, 3 = 3 horizontal balls, and 4 = 3 vertical balls)

(a)

(b)

(c)

(d)

Fig. 20.10. (a) Successful placement: the ball remains in position; (b) Placement error: the ball is released, but its equilibrium position is higher than the gripper; (c) Placement error: the ball is not released and sticks to the gripper; (d) Placement error: the gripper touches the cage (Reprinted with permission from [112]. Copyright 2006 Institute of Physics.)

Concerning the compacity, let us deﬁne it as the following ratio: C = 4π

A , P2

(20.9)

so that the compacity of a disk is equal to 1. According to the results shown in Fig. 20.9, the best strategy seems to use the compacity as an indicator, since it is the largest in the case of a successful picking (1 ball) on the one hand and because it is size independent on the other hand (there is no need to calibrate the camera). These results suggest that the picking task can be considered as successful when the compacity is larger than 0.7.

20.3 Placing Several release strategies have been studied in the literature (a state of the art can be found in [109]). The chosen strategy described below is inspired by the so-called scraping release. Figure 20.10 illustrates several placement errors. The placement reliability has been studied as a function of the ratio between the height h between the gripper and the bearing on the one hand

20.4 Compliance Eﬀect

205

Gripper L

Gripper D

Gripper H

Gripper J

Reliability as a function of the normalized gap 1 0.5 0

0

0.1

0.2

0.3

0.4

0.5

1 0.5 0

0

0.1

0.2

0.3

0.4

0.5

0

0.1

0.2

0.3

0.4

0.5

1 0.5 0 1

Succeeded Failed

0.5 0

0

0.1

0.2

0.3

0.4

0.5

Fig. 20.11. Reliability of the placement as a function of the nondimensional gap between the gripper and the bearing – Gripper J: 0.3 mm, coated; gripper H: 0.5 mm, noncoated; gripper D: 0.5 mm, coated; gripper L: 0.3 mm, noncoated (Reprinted with permission from [112]. Copyright 2006 Institute of Physics.)

(a)

(b)

(c)

(d)

(e)

(f)

(g)

Fig. 20.12. Compliance of the surface tension gripping

and the gripper diameter d on the other hand. The use of this nondimensional ratio allows the comparison between the Φ0.3 mm and the Φ0.5 mm grippers. In Fig. 20.11, the reliability is calculated as the number of successes divided by 10 (each placement has been replicated 10 times). It is represented as a function of the ratio h/d. For gaps smaller than 0.2 times the ball diameter, the placement is always successful.

20.4 Compliance Eﬀect The liquid meniscus provides a lateral compliance, which could be used to balance positioning inaccuracies. This eﬀect must be very dependent on the volume of liquid. Since this latter could not be measured yet, this inﬂuence is still to be studied. The volume of liquid might be measured with ﬂuorescein or its inﬂuence studied by simulation, using the Surface Evolver freeware since the conﬁguration is not axially symmetric (by deﬁnition).

AQ: Citation of Fig. 20.12 is missing.

206

20 Example 4: Application to the Watch Bearing Case Study: Results

(a)

(b)

(c)

Fig. 20.13. Parameters to be studied experimentally. (a) The prestressing ξ applied onto the ball before picking; (b) the waiting time τ during which the prestressing is applied; (c) the picking speed v

20.5 Force Measurement 20.5.1 Introduction The force measurement is necessary to validate the force models described in Chap. 10 and to study experimentally the inﬂuence of nonmodeled parameters such as the prestressing ξ applied onto the ball before picking, the waiting time τ during which this prestressing is applied, and the picking speed v. These parameters are illustrated in Fig. 20.13. 20.5.2 Modiﬁcation of the Force Measurement Test Bed This ﬁrst contact discards the main part of liquid so that the volume of liquid becomes smaller than the conical cavity of the grippers (henceforth, it cannot be seen with the camera). Then, the gripper is applied onto the ball n times without reﬁlling it and the capillary force between the gripper and the ball is measured. After each rupture of the meniscus (i.e., after each contact), a bit of liquid is left on the ball, so that the volume of liquid involved in the manipulation is decreasing (however, it cannot be measured). 20.5.3 Comparison Between Models and Experiments The ﬁrst presented result is the comparison between models of Sect. 10.3 and Sect. 10.4 and the experimental measured force. From Fig. 10.14, two regimes can be distinguished: for β < α (here α = 20◦ ), the force is increasing with β while the volume of liquid remains quite constant. In the case β > α, the volume of liquid quickly increases with the ﬁlling angle while the force is decreasing. The ﬁrst regime is not accessible at all, due to the dramatic sensitivity of the force to the volume of liquid. The second regime encounters severe limitations because the gripper is not optically transparent, hence the volume of liquid cannot be accurately measured (this is still work in progress).

20.5 Force Measurement

207

x 10−4 Simulation result (H)

Force (N)

1.5

Gripper H

Simulation result (J) 1 Gripper J Gripper B 0.5

Gripper F Weight of a 0.5mm diameter ZrO2 ball

0

0

20

40

60

80

100

120

Experiment number

Fig. 20.14. Force measurements with the oil L23: The behavior of grippers J (•) and H (2) is to be compared with simulation in the case of maximal ﬁlling angle β. These simulation results only provide a comparison of the order of magnitude since the volume of liquid is not accurately known (gripper H: 0.5 mm without channel, noncoated; gripper J: 0.3 mm without channel, coated; gripper B (◦): 0.5 mm with channel, coated; gripper F (∆): 0.5 mm with channel, noncoated). For grippers B and F , the presence of an internal channel seems to reduce and stabilize the force (Reprinted with permission from [112]. Copyright 2006 Institute of Physics.)

Therefore, to assess at least the order of magnitude of the force, the simulation has been run until the liquid overﬂows, i.e., when β ≈ 60◦ . The underlying justiﬁcation is the assumption that if the contact between the gripper and the ball is repeated without reﬁlling the gripper, the amount of liquid involved in the micromanipulation is assumed to decrease, hence the force increases. If the reasoning is valid, the simulated force value should be a kind of limit toward the experimental force should tend to. This trend is observed in Fig. 20.14 for grippers H and J, which have no internal channel. For grippers B and F (with an internal channel), the simulation cannot be run because the eﬀects of the channel have not been modeled. Nevertheless, the presence of this channel seems to decrease the force (the force generated by larger grippers B and F is even smaller than the one generated by the smaller gripper J). The conclusions at this stage are that the order of magnitude predicted by the simulation is good but a ﬁner analysis should be done with a transparent gripper in order to measure both the volume of liquid and the force at the same time. Deeper investigations could also be done on the role played by the channel; however, its presence increases the complexity of manufacturing and might be unnecessary for application purpose. Finally, since the force limit does not seem to have been reached in Fig. 20.14, a second set of experiments has been done with a linear motor instead of a manual stage to ensure the gripper motion.

208

20 Example 4: Application to the Watch Bearing Case Study: Results

20.5.4 Ongoing Experimental Study As already stated in the introduction, the inﬂuence of the following parameters on the capillary force FC and the reliability of the process are to be studied: the volume of liquid V , the prestressing ξ applied by the gripper on the component before the picking step, the duration τ of this prestressing (also called delay in what follows), the picking velocity v of the gripper, and the number of repetitions n of the picking task. The following procedure has been applied: 1. Adding liquid (L23 oil, surface tension = 33.5 mN/m, dynamic viscosity = 21.7 mPa s, density = 906 kg/m3 , at 20◦ C) on the gripper 2. Moving the gripper on the ball and applying a prestressing of ξ mN 3. Waiting τ ms 4. Moving the gripper up at constant velocity v In this procedure, attention should be paid to the fact that the picking task is repeated n times (steps 2–4 are repeated n times) on the same ball, introducing artifact as far as the volume of liquid is concerned. Indeed, before the ﬁrst contact between the gripper and the ball, the volume of liquid is only on the gripper side and it is larger than the available volume that can be ﬁlled between the gripper and the ball. Consequently, during the ﬁrst picking (n = 1), some liquid ﬂow from the ball to the beam, leading to the decrease of the actual volume of liquid involved in the manipulation. After some repetitions, however, the system reaches a steady state since there is no leakage of liquid any more (the beam becomes saturated and a nonvolatile liquid has been used – silicone oil L23). For each set of input parameters (ξ, τ , v), the picking force FC is measured n times during n repetitions (without reﬁlling the system with liquid). A typical evolution of FC is shown in Fig. 20.15 (n = 10,000). In a ﬁrst step, the curve grows up, reaches a maximum, and begins to decrease. This is explained in [112] by the fact that by varying the volume of liquid (cf. the experimental procedure which is described above), the force increases, reaches its maximum, and decreases. After that, the capillary force should stabilize: the liquid should stop ﬂooding over the ball and the volume would remain constant. However, the experiments have not been achieved as much as 10,000 times. In the following, attention will only be paid to the increasing part of the curve (most of the experiments contain only 1,000 repetitions) and the experimental data will be ﬁtted with the empirical curve FC = A − B e−Cn by the least squares method (line in the lower part of Fig. 20.15). Consequently, it will be focused on the inﬂuence of the liquid feeding and of the eﬀects of ξ, τ , and v on the values of A, B, and C. The coeﬃcient A represents the maximum intensity of the capillary force, A − B is the capillary force developed with initial volume of liquid and C describes how the capillary force evolves with the samples n.

20.6 Conclusions

209

Capillary Force [mN]

0.1 0.05 0

0

2000

4000

6000

8000

10000

300

400

500

n 0.12

0.1 A A–B 0.08

0

100

200 n

Fig. 20.15. Measure of capillary forces in an experiment of n = 10,000 repetitions, with a prestressing ξ = 224 µN, a delay τ = 500 ms, and a picking speed v = 0.2 mm/s. The bottom subﬁgure shows the ﬁrst 500 repetitions that have been ﬁtted with a curve FC = A − B e−Cn by the least squares method. Note that the lengths A and A − B must be taken from the origin. For sake of clarity C has not been represented (Reprinted with permission from [173], Copyright IWMF 2006.)

20.6 Conclusions Following conclusions can be drawn from the above mentioned results. When the picking error consists in an error centering or in picking up more than one ball, the gripper can be dipped to liquid in order to release the balls. Then, the picking task has to be performed again. When no ball can be picked up, we suggest to tune the volume of liquid, but when the gripper is centered above a ball without neighbors, this latter can always be picked up. We proposed a vision control resting on a blob analysis and a compacity analysis in order to automatically detect the error type. The placing errors have been studied as a function of the gap between the gripper and the bearing. When this gap is smaller than 1/5 of the gripper diameter, the placing task is always successful. Finally, the force results show that the generated capillary force (of the order of 100 µN) is much larger than the weight (of the order of 4 µN). Concerning the liquid supply of the gripper, it can be deﬁnitively concluded that the internal channel through which a pressure is applied is not a good solution. Indeed, the manufacturing is more complicated, the liquid feeding is subject to instabilities, and moreover, the generated force is smaller. Therefore, the tip dipping strategy seems to be the good one. Nevertheless, the validation of the proposed model has only been achieved as far as the order of magnitude of the force is concerned. A more deﬁnitive

210

20 Example 4: Application to the Watch Bearing Case Study: Results

validation will be done as soon as the volume of liquid involved in the micromanipulation will be measured. The repetitions of up to 10,000 picking operations without reﬁlling the gripper seems to be promising, but additional work will be done in a conﬁguration which might be more realistic, i.e., where the ball would be dried between two picking in order to better control the volume of liquid forming the meniscus.

21 Conclusions

21.1 Introduction To conclude this part, we propose to summarize the obtained results in terms of picking on the one hand and of releasing on the other hand. Additionally, some rules will be proposed to be applied to the design of a surface tension based gripper.

21.2 Picking Operations In terms of picking task, and especially in terms of picking force, the inﬂuence of the following parameters can be pointed out: 1. Inﬂuence of contact angles (Figs. 18.13 and 18.14): contact angle θ1 depends on the combination component material-gripping liquid while θ2 depends on the combination gripping liquid-gripper material. Therefore, if the component is imposed, the gripping liquid must be chosen to cause small θ1 , in order to increase the force (it has been shown that the force decreases as a function of θ1 , a little more slowly than the cosine function). Note that for a given component material, a low energy liquid (i.e., with small γ) provokes smaller θ1 but is less eﬃcient as the force is also proportional to γ. Therefore a compromise must be found. Additionally, it must be noted that in the case of the conforming gripper used in the watch bearing case study, the force is attractive even for nonwetting coating, as indicated in Fig. 10.12. 2. Inﬂuence of volume of liquid (Fig. 18.9): for spherical grippers, the volume of liquid plays almost no role. Consequently the dispensing accuracy or evaporation, for example, are not parameters of the utmost importance here. On a diﬀerent way, the force can remain quite steady if the amount of liquid can be kept constant, which is more or less the case with a foam gripper (Fig. 18.26). Nevertheless, for conical grippers, the larger

212

21 Conclusions

the volume, the larger the force. At the contrary, in the case of the conforming gripper of the watch bearing case study, the force increases with decrease in volume of liquid (Fig. 10.14), which is of the utmost interest in a high throughput application. Consequently, no general rule can be drawn concerning the inﬂuence of the volume of liquid: it depends on the conformation between the component and the gripper geometries. Moreover, the amount of liquid also determines the approaching contact distance and the rupture distance. 3. Inﬂuence of the gripper’s geometry: cylindrical grippers lead to the largest forces. As far as spherical grippers are concerned, the force is proportional to their radius as illustrated by the Israelachvili approximation [89]: F ≈ 4πRγ cos θ.

(21.1)

For conical grippers (Fig. 18.9) with silicone oil R47V501 , the developed force FR47V50 is approximatively given by log10

FR47V50 p 4 ≈ − log10 , Fo 3 po

(21.2)

where po = 10◦ , p is the angular aperture (in ◦ ) of the considered conical gripper, and Fo (in N) is given by log10

Fo V ≈ 0.33 log10 −9 , −3 0.87 × 10 N 10 L

(21.3)

where V is the volume of silicone oil (in L). The constants 0.87 × 10−3 N, 10−9 L, 0.33, and − 43 have been read in Fig. 18.9. Note that for other liquids (i.e., other surface tensions γ), the force can be assumed to be proportional to γ, and therefore F FR47V 50

=

γ γR47V 50

.

(21.4)

This decrease of the force as a function of the angular aperture has to be taken into account when considering the manufacturing accuracy of the cone. 4. Inﬂuence of surface tension (Figs. 18.10 and 18.11): the force developed by a liquid bridge is proportional to surface tension. Nevertheless, liquids with a low surface tension better wet solids, leading to smaller contact angles; therefore, a compromise has to be found. 5. Inﬂuence of the gap (Fig. 18.7): the force decreases with the separation distance so that the maximal force is observed when gripper and component touch each other. 6. As an order of magnitude, forces up to several milli-Newton can be reached easily (1 mN corresponds to a mass m = 100 mg). The rule of thumb is 1 mN for a 1 mm sized component. 1

γ = 20.9 mN m−1 .

21.3 Releasing Strategies

213

We have seen in Sect. 20.2 that in case of picking more than one component at a time, the gripper could be cleaned by dipping it to a liquid tank. Of course, a better solution would be to develop a speciﬁc feeder according to the application. This will be considered in the perspectives of this work, at least as far as the watch bearing is concerned.

21.3 Releasing Strategies In a micro driven design, care should also be taken of the possibilities to release the component. In macromanipulation, indeed, the picking force Fmax is replaced by Fmin = 0 at release while in the case of a surface tension based gripper (or any adhesive principle based gripper), there is always a residual adhesion force Fmin > 0 that can prevent the component from being released if Fmin is larger than its weight W . Consequently, an adhesion or surface tension based gripper can only be designed for a restricted mass range [mmin , mmax ] of components as illustrated in Table 21.1. To compare the diﬀerent release strategies in a quantiﬁed way, the following “adhesion” ratio has been introduced: φ=

Fmin , Fmax

(21.5)

φ ranges from 0 (no residual adhesion, the releasing task is not a problem) to 1 (residual adhesion force is as large as the picking one, handling cannot be performed). Let us now summarize the diﬀerent release strategies (sorted by decreasing φ): 1. Reduce the volume of the liquid bridge (Fig. 18.9): this strategy cannot be used with spherical grippers since the developed force is almost independent from the amount of liquid but can be achieved for conical grippers. Figure 18.9 indicates that if the volume is reduced 10 times, the force is reduced by 1.541 (log F − log Fo = 0.432): φ ≈ 0.649.

(21.6)

2. Tilt the gripper with respect to the component (Fig. 18.15): if the orientation of the gripper is changed from 0 to 5◦ , the ratio is φ≈

2.4 mN ≈ 0.436 5.5 mN

(21.7)

Table 21.1. Mass range Mass position m > mmax mmax > m > mmin mmin > m

Description Component cannot be picked up Component can be picked up and released Component can be picked up but cannot be released

214

21 Conclusions

and, if the gripper is tilted to 10◦ φ≈

1.2 mN ≈ 0.218. 5.5 mN

(21.8)

3. Move the gripper in the shear direction: in this case, if the component is blocked laterally, the gripper can be moved in the xy plane (i.e., perpendicular to the axial symmetry axis z) and the force to generate depends on the friction between the gripper and the component: Fshear = fo F,

(21.9)

where fo is the static friction coeﬃcient (“static” because the gripper is moved from rest). In this case, the ratio of the forces to develop is simply given by fo . This release is sometimes called scrapping release method [50]. This method has been successfully implemented in the watch bearing case study. Note that in order to achieve a reliable release of a spherical component, the gap between the latter and the summit of the cavity should not be larger than a ﬁfth of the ball diameter (Fig. 20.11). 4. Increase the gap (Fig. 18.7): by increasing the gap, the force can be reduced eﬃciently. In the example of Fig. 18.7, it can be seen that a gap z = 500 µm leads to φ≈

0.10 mN ≈ 0.105. 0.95 mN

(21.10)

5. Change the gripper geometry (Fig. 18.9): if the angular aperture was increased from 0◦ to 5◦ , the force reduction would lead to φ≈

2 mN ≈ 0.100. 20 mN

(21.11)

6. Use dynamical eﬀects (Sect. 18.10): let us assume that the picking task of a mass m is achieved with a zero velocity and a force F . Therefore, the limit for m is Fg (g = 9.81 m s−1 ). If now an acceleration a is imposed to the gripper, the heaviest acceptable mass is m = m g/a + g. Therefore, the force seems to have been reduced by g/a + g. If a ≈ 100 m s−2 , the ratio becomes g ≈ 0.089. (21.12) φ= a+g Note that this method has to be envisaged carefully because of its low positioning accuracy. 7. Use an auxiliary releasing tip (Fig. 18.17): for example, by substituting an auxiliary PTFE tip to a steel cylindrical gripper generating forces about 5 mN, the force can be reduced to about 60 µm, leading to φ≈

0.060 mN ≈ 0.012. 5 mN

(21.13)

21.4 Design Aspects

215

8. Control the contact angle (for example, using electrowetting): if the contact angle was controlled and set to π2 , the residual force could be avoided as it essentially depends on cos θ. In this case φ ≈ 0.

(21.14)

Several strategies can be combined, for example, a gripper tilting achieved with a dynamical release: in this case the total eﬃciency of the release strategy can be assessed by Φ=

n !

φi .

(21.15)

i=1

21.4 Design Aspects In a given application, the component is imposed, that is its material and its geometry cannot be changed. The freedom of the designer concerns the handling liquid (surface tension, volatility, and dynamic viscosity), the gripper (material and coating, geometry), and the releasing strategy. •

•

•

•

Surface tension: The larger the surface tension, the larger the capillary force. Nevertheless, a liquid with a lower surface tension will better wet the component, leading to an opposite eﬀect. This issue must be carefully addressed. Volatility: To achieve experimental measurements, we recommend to use a nonvolatile liquid, such as, for example, the silicone oil we have used in this work. Since it does not evaporate, the volume of liquid remains constant during the experiment. Nevertheless, this can be a drawback from the application point of view, since a nonvolatile liquid will leave residual traces on the component. If the assembly task can be achieved very quickly (less than 1 sec to ﬁx the order of magnitude), volatile liquids such as alcohols are suitable: they do not leave traces. Contact Angle Hysteresis: To achieve experimental measurements, it is of the utmost importance to know whether the advancing or the receding contact angle should be used in the models. Moreover, the larger the hysteresis, the larger the variability of the results. Consequently, water is not recommended. At the contrary, silicone oils or alcohols seem to be better; however, it is recommended to ﬁrst measure the contact angles with the liquid one proposes to use. Dynamic viscosity: The proposed force models rely on an equilibrium assumption. Therefore, the viscosity does not play any role in these models (i.e., the viscosity does not change the amount of force at equilibrium). Nevertheless, in high throughput machines, the transient eﬀects become dominant. We have proposed an approximation of the characteristic damping time τ by solving the Lucas-Washburn equation:

216

21 Conclusions

τ=

•

•

•

ρd2 1 = . λ 6µ

(21.16)

Implementing a dynamical simulation is one of the perspectives of this work. Gripper Material: A gripper with high energy (metals for example) leads to smaller contact angles θ2 than a gripper with low energy (polymers for example). Nevertheless, the results of Fig. 10.12 indicate that a nonwetting gripper can also lead to an attractive capillary force. The choice of the material can also be ruled by manufacturing aspects. Gripper Surface: According to the Wenzel model (11.5), the roughness ampliﬁes the wetting behavior: angles lower than 90◦ are decreased by roughness, while the angle increases if it is larger than 90◦ . This means that the gripping force of a capillary gripper could be increased by using rough gripper tips. The surface impurities increase the contact angle hysteresis (Fig. 11.1). Finally, we have observed that the presence of a coating could be a drawback if it not mechanically resistant to wear (the gripper is subject to many contacts). Of course, a wear resistant coating permits to change the contact angle θ2 . Gripper Geometry: The geometry of the gripper is of the utmost importance. Actually, the important parameter to keep in mind is that near contact, the “pressure” or “Laplace” term of the capillary force dominates the “tension” one (see Fig. 18.8). To maximize the capillary force, the gripper should maximize the pressure diﬀerence pin − pout and also the area on which this pressure diﬀerence is acting. The ﬁrst objective can be done with small θ1 and θ2 but also by keeping distance between the gripper and the component as small as possible, i.e., designing a conforming geometry. This also respects the second condition since, for a given volume of liquid, a conforming gripper leads to a larger contact circle radius, henceforth to a larger area. This principle is qualitatively discussed from Fig. 21.1. We see on this picture that for the (hypothetic) depicted gripper, the distance between the gripper and the component is smaller for a volume of liquid corresponding to a meniscus wetting in B than in A. Hence, the pressure diﬀerence ∆pB is larger than the pressure diﬀerence ∆pA . Moreover, since the meniscus radius rB is larger than the meniscus radius rA , the area over which this pressure diﬀerence acts is much larger, leading to a larger force (FB > FA ). If the volume of liquid is increased a bit more, the meniscus will wet the component at a distance rC > rB from the symmetry axis. Therefore, the area is increased, but the pressure diﬀerence is reduced, since the distance between the component and the gripper increases. Both eﬀects ﬁght one another, and qualitatively, the force can increase or decrease, which is shown in the bottom right sketch of Fig. 21.1. We also see on this picture the qualitative evolution of the volume of liquid. It is clear that it will be diﬃcult to master the volume of liquid between VA and VB , since the sensitivity of the volume is very

21.4 Design Aspects

217

gripper

r A

B

C

component

V

F

rA

rB

rC

r

rA

rB

rC

r

Fig. 21.1. Design rules

(a)

(b)

Fig. 21.2. (a) Example of a truncated conical gripper, which allows a large pressure diﬀerence together with a large acting area; (b) Example of the gripper proposed by Schmid et al. [155] (Courtesy of D. Schmid, EPFL/STI/IPR/LPM.)

low. At the contrary, there is a dramatic increase of the volume of liquid as soon as r > rC . These principles can be applied to optimize the design of a gripper intend to pick ﬂat components. We see in Fig. 21.2a that giving the gripper the shape of a truncated cone, we increase the “acting” area2 by keeping a small distance between the gripper and the component, 2

The area on which the pressure diﬀerence acts.

218

•

21 Conclusions

leading to a large pressure diﬀerence. We let the reader compare this conﬁguration with the spherical or the conical gripper presented in a previous chapter. This has been applied in [155] to pick up square molybdenum dies (100 × 100 × 20 µm3 ), simulating optical components. An example of the used gripper is given in Fig. 21.2b. Scaling Law: According to (6.14), the capillary force linearly depends on the system size. Therefore, this principle is of the utmost interest in miniaturization.

22 Conclusions and Perspectives

22.1 Conclusions The goal deﬁned at the very beginning of this work was to study the role of surface forces in microassembly, and particularly, to assess the inﬂuence of capillary forces in one of the assembly tasks, i.e., handling. One of the ﬁrst issues to be addressed was to grasp the framework of microassembly. This has been led in Part I: it is shown that the classical approaches to assemble components can be renewed by using diﬀerent assembly models (such as, for example, internal or stochastic assembly) or diﬀerent handling principles. From a literature review on gripping principles, it turns out that one of these handling principles, namely the surface tension based handling, can lead to the set up of a capillary gripper prototype. Among the advantages of such a gripper, let us cite a large force (several milli-Newton) compared with the weight of small components, a favorable downscaling law (i.e., better than that of the vacuum gripper since the force only decreases as a linear function of the characteristic size), the ability to pick up components that present only one accessible surface, a picking leading to lower contact force than those exerted by a vacuum gripper. However, several points must be assessed and mastered: is this kind of grippers suitable for use in high acceleration assembly machines? In this case, how can the release task be achieved? What is the residual volume of liquid left on the component after it has been released? What are the design parameters of such a gripper? To answer these issues, the modeling of capillary forces in Part II has been adapted to the case study of a gripping task and a numerical code has been written in order to simulate the working of the capillary handling according to the following parameters: surface tension and volume of the used liquid (i.e., water and silicone oil), shape, material and size of the gripper (several conical and spherical grippers made of steel or PTFE), material of the component (steel, silicon), and kinematics of the gripper (traveling range and acceleration). These models output the meniscus shape, the capillary force it exerts, and the critical distance at which the meniscus will break. Two examples are

222

22 Conclusions and Perspectives

developed: ﬁrst, these models are applied to the case study of a watch ball bearing with 0.3 and 0.5 mm diameter zirconium balls, and second, the numerical implementation of the double iterative scheme presented in Chap. 8 is exhaustively detailed. Part III is devoted to the experimental aspects. A test bed has been set up (chapter 17) in order to validate the simulation code and also to get information being outside the axially symmetric assumptions framework of the simulation (for example, to study the inﬂuence of tilting the gripper with respect to the component). From these simulations and experiments, it turns out that capillary forces are large enough to be used as a gripping principle in microassembly (even in automated assembly machines with high accelerations): they can actually pick up millimetric components with accelerations up to 100 ms−2 (10 G). The release capability has been formalized through a so-called “adhesion” ratio φ that has been quantiﬁed for several release strategies in Chap. 21: most promising ones probably consist in tilting the gripper, moving the gripper laterally, increasing the gap, changing the gripper geometry, using an auxiliary releasing tip, or controlling the contact angle. Dynamical release has been rejected because of its poor positioning accuracy. As the presence of a residual volume of liquid on the component after releasing is unavoidable, an additional model has been adapted in order to evaluate it. Moreover, it has been shown that this residual volume could disappear in a few seconds due to evaporation (at least for water at ambient temperature). A list of design parameters has been established and the inﬂuence of each of them has been quantitatively assessed. It has been shown that the developed capillary force was proportional to surface tension γ, that the inﬂuence of the volume of liquid would depend on the kind of gripper geometry (the force does not depend on it in the case of a sphere–plane interaction, increases with increasing volume in the case of the cone–plane interaction, decreases with increasing volume in the case of the conforming watch bearing related gripper); the inﬂuence of the gripper size and that of the contact angles have been described by mathematical laws (Chap. 21). This third part ends with design suggestions. An originality of this work lies in the classiﬁcation of gripping principles according to the way they tackle surface forces issues: it led to the proposal to use capillary force as a gripping principle (even if it had already been partially and independently suggested by two German researchers [13] and [72]). The main contributions of this work are the transfer and the adaptation of scientiﬁc knowledge from the chemists and physicists community to the microassembly community. The main results of this book are the quantiﬁed comparisons of picking strategies on the one hand and release strategies on the other hand.

22.2 Perspectives

223

22.2 Perspectives The perspectives of this work are manyfold. First, in terms of modeling, two topics emerge from this work: the capillary condensation and the dynamical simulation. The capillary condensation problem is the companion topic of surface tension gripping. According to the Kelvin equation (11.7), the humidity of the surrounding environment condensates in some conditions, leading to the formation of a liquid bridge at the nanometric scale, and henceforth to adhesive capillary forces, which are assumed to be responsible for MEMS breakdown and for a part of the pull-oﬀ in atomic force microscopy. An introduction to this problem can be found in [34]. The dynamical simulation of the meniscus evolution would allow to better understand the transient phase and will probably of importance to model the surface tension based gripping in high throughput assembly machines, or at least as soon as the cycle time of the assembly process will become of the order of magnitude of the characteristic damping time (about 10 ms for water). Second, in terms of materials, a challenging topic is the active control of the wetting properties. This would allow to switch on and oﬀ a surface tension based gripper, making the adhesion ratio very small. This idea is already approached with electrowetting [129] (i.e., control of the apparent contact between a conductive substrate and a conductive droplet by applying a voltage between the substrate and the droplet. To avoid the short-circuit, the substrate is coated with a nonwetting and nonconductive coating) and opto-electrowetting [37] (the impedance variation acting in the electrowetting is controlled optically). Third, in terms of technology, the surface tension based gripper should be coupled with a dedicated powerful feeder, able to separate, orientate, and position the components. This cannot be developed independently of the application, so that the next step of this work will be the application of these principles to an industrial case study as, for example, the assembly of SMD components. An additional investigation track is the coupling between an adhesive principle (surface tension eﬀect) and a repulsive principle (ultrasonic vibrations) in order to achieve a controllable noncontact gripping.

A Modeling Complements

This appendix aims at detailing the mathematical developments required to calculate the analytical approximations of the capillary forces, based on energetic approach (Chap. 7), and to establish the expression of the interfacial energies (L–V and total energies) used in Chap. 13 to study the rupture conditions and the related volume repartition. It also illustrates the relation between the capillary force and the interfacial energy, to illustrate the correctness of the interfacial energy expression we used in this appendix.

A.1 Analytical Approximations of the Capillary Forces A.1.1 Preliminary 1. Deﬁnitions

1 2π 3 3 A(φ) ≡ 1 − cos φ + cos φ , 3 2 2 dA = π sin3 φ. dφ 2. Properties φ2 φ4 + + O(φ6 ), 2 24 φ4 cos2 φ = 1 − φ2 + + O(φ6 ), 3 3 7 cos3 φ = 1 − φ2 + φ4 + O(φ6 ), 2 8 φ3 sin φ = φ − + O(φ5 ), 6 φ4 sin2 φ = φ2 − + O(φ6 ), 3 cos φ = 1 −

228

A Modeling Complements

sin3 φ = φ3 + O(φ5 ), π A(φ) = φ4 + O(φ6 ), 4 dA = πφ3 + O(φ5 ), dφ 1 − cos φ ≈

φ2 sin φ2 ≈ . 2 2

A.1.2 Between a Sphere and a Plane The force between a sphere and a plane is developed in [89]. The used notations are deﬁned in Fig. A.1. In this ﬁgure, φ0 and r0 are arbitrary constants. Their exact value does not play any role because the force will be calculated from the diﬀerentiation of the interfacial energy W with respect to the gap z between the sphere and the plane [89]: F =−

dW . dz

(A.1)

Let us write the interfacial energy of the system: W (z) = ASL γSL + ASV γSV + Σγ = γSL πr2 + γSV π(r02 − r2 ) + γ2πr z + R(1 − cos φ) (A.2) 2 2 +γSL 2πR (1 − cos φ) + γSV 2πR (1 − cos φ0 ) − (1 − cos φ) . Since φ is assumed to be small, W can be rewritten as: W (z) = πr2 (γSL − γSV ) + γ2πrz + γπrR sin2 φ + γSV πr02 +πR2 sin2 φ(γSL − γSV ) + γSV πR2 sin2 φ0 and, by considering the Young–Dupr´e equation (γ cos θ = −γSL + γSV ):

R φ

φ0

h z r r0

Fig. A.1. Studied conﬁguration

A.1 Analytical Approximations of the Capillary Forces

229

W = −2πR2 sin2 φγ cos θ+γSV πr02 +γ2πrz +γπR2 sin3 φ+γSV πR2 sin2 φ0 . (A.3) Let us now consider the derivative of W : dW dφ dφ = −4πR2 sin φ cos φγ cos θ + γ2πR sin φ + γ2πzR cos φ dz dz dz dφ 2 2 +3γπR sin φ cos φ dz or, by assuming sin φ ≈ φ and cos φ ≈ 1: dW dφ dφ dφ = −4πR2 φγ cos θ + γ2πRφ + γ2πRz + 3γπR2 φ2 . dz dz dz dz

(A.4)

(A.5)

The value of dφ/dz must be evaluated in (A.5). Therefore, the meniscus volume is assumed to be constant, leading to dV /dz = 0. Moreover, the meniscus will be assumed to be cylindrically shaped so that the volume is the diﬀerence between the external liquid cylinder and the volume of the spherical cap inside the external cylinder: 2πR3 3 cos3 φ 1 − cos φ + . V = πr2 z + R(1 − cos φ) − 3 2 2

(A.6)

Once again the assumption of small φ is made, leading to the following approximation: 3 cos3 φ 2πR3 πR3 4 1 − cos φ + = A(φ)R3 ≈ φ . 3 2 2 4

(A.7)

The ﬁnal expression for V is now given by: πr2 R πR3 4 sin2 φ − φ 2 4 πR3 πR3 4 sin4 φ − φ = πR2 sin2 φz + 2 4 so that: dV dφ dφ = 2πR2 z sin φ cos φ + πR2 sin φ + 2πR3 sin3 φ cos φ dz dz dz 3 3 dφ −πR φ dz dφ dφ 2 = 2πR zφ + πR2 φ2 + πR3 φ3 dz dz =0 −πR2 φ2 dφ = ⇒ 2 dz 2πR φz + πR3 φ3 −1 = 2z . φ + Rφ V = πr2 z +

(A.8) (A.9)

(A.10)

(A.11)

230

A Modeling Complements

The total capillary force is then given by substituting this latter result into (A.5): F =−

4πR2 φγ cos θ γ2πRz 3γπR2 φ2 − γ2πRφ + 2z + 2z . 2z φ + Rφ φ + Rφ φ + Rφ

Since h = R(1 − cos φ) ≈

R 2

sin2 φ ≈

(A.12)

R 2 2φ :

F =−

4πRγ cos θ γ2πRz 3γπRφ − γ2πRφ + 2z + 2z 2z + 1 + Rφ 2 Rφ φ Rφ2 + 1

=−

3γπRφ γ2πRz 4πRγ cos θ + z − γ2πRφ + 2z . z h +1 h +1 φ + Rφ

The last three terms of (A.13) represent the contribution of the “LV” interface to the total interfacial energy. Let us assess their relative importance with respect to the ﬁrst term. Their sum is given by: πRγφ(Rφ2 − 2z) . Rφ2 + 2z

(A.13)

The ratio of the the ﬁrst term to the sum of the last three ones is equal to: 4πRγ cos θ z h +1 πRγφ(Rφ2 −2z) Rφ2 +2z

=

4 cos θh . φ(h − z)

(A.14)

If z = 0, this ratio tends toward inﬁnity if φ tends to zero. Since φ cannot be exactly equal to zero, the last three terms can be neglected with the (now) classical assumption of small φ (φ ≈ sin φ). This leads to the well-known approximation [89]: Fmax = −4πRγ cos θ.

(A.15)

If z = 0 but by neglecting the contribution of lateral area to W , the total capillary force can be rewritten as follows: F =−

4πRγ cos θ . z h +1

(A.16)

A.1.3 Between Two Spheres Let us assume two spheres S1 and S2 characterized by their radius R1 and R2 such as depicted in Fig. A.2.

A.1 Analytical Approximations of the Capillary Forces

φ2

231

R2 SV2

SL2

h2 r

LV

z h1

SL1

SV1

φ1

R1

Fig. A.2. Notations

Preliminaries

≡ R1 sin φ1 ≡ R2 sin φ2 , dφ1 dφ2 = R1 cos φ1 = R2 cos φ2 , dz dz ≈ R 1 φ1 ≈ R 2 φ2 , dφ1 dφ2 ≈ R1 = R2 , dz dz dφi , φi ≡ dz dr ≈ R1 φ1 ≈ R2 φ2 . r ≡ dz

r dr dz r dr dz

Expression of the Interfacial Energy The total interfacial energy W can be expressed as follows: W = γ2πr(z + R1 (1 − cos φ1 ) + R2 (1 − cos φ2 )) + γSL2 2πR22 (1 − cos φ2 ) +γSV2 2πR22 (C − (1 − cos φ2 )) + γSL1 2πR12 (1 − cos φ1 ) +γSV2 2πR12 (C − (1 − cos φ1 )),

(A.17)

where C and C are arbitrary constants. With the assumptions that:

232

A Modeling Complements

C = C = 0, φ → 0 ⇒ sin2 φ ≈ 2(1 − cos φ), r = R1 sin φ1 = R2 sin φ2 , γSVi = γSLi + γ cos θi . W can be rewritten into: W = πrγ(2z + R1 sin2 φ1 + R2 sin2 φ2 ) + πγSL2 R22 sin2 φ2 −γSV2 πR22 sin2 φ2 + πγSL1 R12 sin2 φ1 − γSV1 πR12 sin2 φ1 = 2πrγz + πγR12 sin3 φ1 + πγR22 sin3 φ2 − πR22 sin2 φ2 γ cos θ2 −πR12 sin2 φ1 γ cos θ1 ⇒

(A.18)

dW = 2πγr + 2πγzr + 3πγR12 sin2 φ1 cos φ1 φ1 dz +3πγR22 sin2 φ2 cos φ2 φ2 − 2πγR22 sin φ2 cos φ2 φ2 cos θ2 −2πγR12 sin φ1 cos φ1 φ1 cos θ1 .

(A.19)

As it can still be assumed that φ → 0, sin φ ≈ φ and cos φ ≈ 1, leading to: dW = 2πγr + 2πγzr + 3πγR12 φ21 φ1 + 3πγR22 φ22 φ2 − 2πγR22 φ2 φ2 cos θ2 dz −2πγR12 φ1 φ1 cos θ1 (A.20) (A.21) It can be emphasized that r ≈ R1 φ1 ≈ R2 φ2 and that consequently, φ2 = φ1 (R1 /R2 ). φ1 must still be determined, so the conservation of liquid volume is assumed (dV /dz = 0). The volume of liquid can be approached by the volume of the cylinder of radius r and height z + h1 + h2 decreased by the volumes of two spherical of radii R1 and R2 , and limited by the angles φ1 and φ2 : V = πr2 (z + R1 (1 − cos φ1 ) + R2 (1 − cos φ2 )) − A1 R13 − A2 R23 leading to: dV = 2πrr (z + R1 (1 − cos φ1 ) + R2 (1 − cos φ2 )) dz +πr2 (1 + R1 φ1 φ1 + R2 φ2 φ2 ) − πφ31 φ1 R13 − πφ32 φ2 R23 ≡0 r2 2rz + rR1 φ21 + rR2 φ22 + r2 φ1 + r2 φ2 − φ31 R12 − R22 φ32 r =− 2z + rφ1 + rφ2 r =− . (A.22) 2z + r2 ( R11 + R12 )

⇔ R1 φ1 = −

A.2 Volume Repartition by the Energetic Approach

233

Equation (A.21) can now be rewritten into: dW πγr = dz 2z + r2 ( R11 +

1 (2z R2 )

− R1 φ21 − R2 φ22 + 2r(cos θ1 + cos θ2 )).

At contact (z = 0), the assumption of small φi leads to: dW πγr = 2 1 dz r ( R1 +

1 2r(cos θ1 R2 )

+ cos θ2 ).

Let us note 2 cos θ ≡ cos θ1 + cos θ2 and 1/R ≡ 1/R1 + 1/R2 so that the last equation ﬁnally leads to: F ≡−

dW = −4πγR cos θ. dz

(A.23)

It can then be concluded that at contact and with small amounts of liquid (φi ≈ sin φi ), the force between two spheres with radii R1 and R2 is equal to that between a plane and a sphere of radius R given by 1/R = 1/R1 + 1/R2 .

A.2 Volume Repartition by the Energetic Approach Let us assume an “analytical” conﬁguration, i.e., a cylindrical gripper parallel to the component. A.2.1 Assumptions, Notations, and Mathematical Preliminaries 1. The volume V is constant – The meniscus is assumed to be of cylindrical shape V = πr2 z – The volume of both spherical caps after rupture is given by: V = V1 + V2 ⇒ dV2 = −dV1 2. Ai ≡ (2π/3)(1 − (3/2) cos θi + (1/2) cos3 θi ) (As Ai R3 represents the volume of a spherical cap with radius R and limited by an angle θi , Ai ≥ 0, and dAi /dθi > 0) 3. αi ≡ (1/6)(2 − cos θi − cos2 θi ) ≥ 0 4. Ai /αi = 2π(1 − cos θi ) 5. K ≡ (A2 /A1 )(α1 /α2 )3 6. K =

A2 A1

7. 2 cos θ ≡ cos θ1R + cos θ2R 8. γSVi = γSLi + γ cos θi 9. θi = contact angle made by the ith droplet, usually the advancing contact angle

234

A Modeling Complements

10. λi ≡

Ai α3i

θi = 72π (2−cos1−cos θi −cos2 θi )2 ⎡ ⎢ dλi sin θi (2 − cos θi − cos2 θi )2 ⎢ sign = sign ⎢72π dθi ⎣ (2 − cos θi − cos2 θi )4 >0

×

⎤

−2(1 − cos θi )(2 − cos θi − cos2 θi )(sin θi + sin 2θi ) ⎥ ⎥ ⎥ 2 4 ⎦ (2 − cos θi − cos θi ) >0

= sign(cos θi + 3 cos θi − 4) = −1 2

(A.24)

11. The angles involved in W (Fig. A.3a) are receding ones, those involved in U (Fig. A.3b) are between the receding and the advancing angles.

A.2.2 L–V Interfacial Energy The energy of the LV interface WLV is given by: WLV = 2πγrz √ √ √ = 2γ π V z. The energy of the two droplets is given by: ULV = 2πγ[R12 (1 − cos θ1 ) + R22 (1 − cos θ2 )] A1 2 A2 2 =γ R + R α1 1 α2 2 1/3 1/3 A1 A2 2/3 2/3 =γ V + V since Vi = Ai Ri3 , α1 1 α2 2

R2 q2 z

z q1 R1

r

(a) Fig. A.3. Notations

(b)

A.2 Volume Repartition by the Energetic Approach

dULV =0 dV1

1/3

1/3

2 A1 2 A2 −1/3 −1/3 =γ V1 − V 3 α1 3 α2 2 3 V2 A2 α1 ⇒ = ≡ K . V1 A1 α2

235

We conclude that the repartition of the volumes that extremes the interfacial energy is given by: V , (A.25) 1 + K KV V2 = . (A.26) 1 + K ∗ of ULV corresponding to this volumes repartition is given by: The value ULV ⎤ ⎡ α21 A2 2/3 1/3 1/3 ( ) 2 1 A A A α2 1 ∗ ⎦ ULV = γ⎣ 1 + 2 α1 (1 + A2 α313 )2/3 α2 (1 + A2 α313 )2/3 V1 =

=γ

A1 α2

2

A1 α22 A2 α1 + α1 α2

A1 α2

V α23 A1 + α13 A2

2/3

∗ Let us show that ULV is a maximum by evaluating the second derivative of ULV with respect to V1 : 1/3 1/3 d2 ULV 2γ A1 A2 −4/3 −4/3 =− V + V dV12 9 α1 1 α2 2 >0 <0 ∗ d is negative because Ai , αi , and Vi are positive. Consequently, ULV is a maximum. As a concluding remark of this subsection, it must be noted that if θ1 > θ2 , λ1 < λ2 , leading to V1 < V2 . Indeed, dλ/dθ < 0 ⇒ λ1 < λ2 , and V2 /V1 = K = λ2 /λ1 . 2

ULV /dV12

A.2.3 Total Interfacial Energy

W = 2πγrz + πr2 γSL1 + πr2 γSL2 − πr2 γSV1 − πr2 γSV2 = 2πγrz − πr2 γ(cos θ1 + cos θ2 ) = 2πγrz − πr2 γ cos θ √ √ √ 2V γ cos θ = 2γ π V z − , z

236

A Modeling Complements

( U = γ 2π R12 (1 − cos θ1 ) + 2π R22 (1 − cos θ2 ) ) −πR12 sin2 θ1 cos θ1 − πR22 sin2 θ2 cos θ2 = 3γA1 R12 + 3γA2 R22 1/3 2/3 1/3 2/3 = γ 3A1 V1 + 3A2 V2 ,

(A.27)

dU =0 dV1

1/3 −1/3 1/3 −1/3 , = γ 2A1 V1 − 2A2 V2

⇒

V2 A2 = ≡ K. V1 A1

It can be pointed out that this last expression can be rewritten as: V2 V1 = ⇔ R1 = R2 . A2 A1 The volumes V1 and V2 are given by: V , 1+K KV V2 = . 1+K The value U ∗ of U corresponding to this volumes repartition is given by: U ∗ = 3γ A1 R12 + A2 R22 . V1 =

As R13 = R23 =

V1 A1

=

V2 A2

=

V A1 +A2 ,

U ∗ can be written as follows:

U ∗ = 3γV 2/3 (A1 + A2 )1/3 . Let us show that U ∗ is a maximum by evaluating the second derivative of U with respect to V1 : d2 U 2γ 1/3 −4/3 1/3 −4/3 = − (A1 V1 +A V ), dV12 3 2 2 >0 <0

d is negative because Ai and Vi are positive. Consequently, U ∗ is a maximum. As a concluding remark of this subsection, it must be noted that if θ1 > θ2 , A1 > A2 , leading to V1 > V2 . 2

U /dV12

B Geometry Complements

This appendix covers several elements. First we remind the lateral area and the volume of a spherical cap (= portion of a sphere): This result is used in a widespread manner throughout this work. Then, elements of diﬀerential geometry are given in order to calculate the mean curvature of an axially symmetric surface (this result is used to get (8.13)). This appendix also gives the equations of the catenary curve, used in the validation of the simulation code (Sect. 18.2.1).

B.1 Area and Volume of a Spherical Cap Let us consider a spherical cap ABC, deﬁned by its radius R and the limiting angle α such as depicted in Fig. B.1. The lateral area and the volume of this spherical cap can be calculated as follows: 2π α dϕ R2 sin θ dθ S(α, R) = 0

0

= 2πR2 (1 − cos α)

(B.1)

and the volume V is given by: 2π V (α, R) =

α dϕ

0

R dθ

0

r2 sin θ dr

r(θ)

1 2πR 3 3 = 1 − cos α + cos α , 3 2 2 3

α where r(θ) = R cos cos θ .

(B.2)

238

B Geometry Complements

B P2

P1 C

A α

θ

r(q)

R O

Fig. B.1. Deﬁnition of the spherical cap

B.2 Diﬀerential Geometry of Surfaces B.2.1 Mean Curvature of a Surface Let S be a surface given by its vectorial equation: S(u, v) ≡ OP = r(u, v) and let us deﬁne the following diﬀerential operator [47]: ⎧ ∂r ∂r E = ∂u . ∂u ⎪ ⎪ ⎪ ∂r ∂r ⎪ ⎪ F = . ⎪ ∂u ∂v ⎪ ⎪ ∂r ∂r ⎨ G = ∂v . ∂v , ∂2r ⎪ L = ∂u2 .1n ⎪ ⎪ ⎪ ⎪ ∂2r ⎪ M = ∂u∂v .1n ⎪ ⎪ ⎩ ∂2r N = ∂v2 .1n

(B.3)

(B.4)

where ¯ 1n is the normal vector deﬁned by: ¯ 1n =

∂ r¯ ∂u ∂ r¯ || ∂u

× ×

∂u ¯ ∂v . ∂u ¯ ∂v ||

(B.5)

Note that the sign of this latter result does not make geometrical sense: The permutation between u and v changes the orientation of ¯1n . According to [47], the curvature Kn of a normal section of this surface is given by:

B.2 Diﬀerential Geometry of Surfaces

Kn =

1 L du2 + 2M du dv + N dv 2 = Rn E du2 + 2F du dv + G dv 2

or with λ ≡ Kn =

239

(B.6)

du dv :

1 Lλ2 + 2M λ + N . = Rn Eλ2 + 2F λ + G

(B.7)

This equation can be rewritten as follows: λ2 (LR − E) + 2λ(M R − F ) + N R − G = 0.

(B.8)

It means that if they exist, there are usually two directions λ1,2 for a given curvature R: F − M R ± (M R − F )2 − (LR − E)(N R − G) . (B.9) λ1,2 = LR − E If the directions λ1 and λ2 are equal, they indicate a principal direction. In this case, the corresponding curvature radius R is a principal curvature radius, given by: (M R − F )2 − (LR − E)(N R − G) = 0

(B.10)

that can be rewritten into: 1 1 (EG − F 2 ) + (2M F − EN − GL) + (LN − M 2 ) = 0. R2 R

(B.11)

This equation has two solutions R1 and R2 whose sum is given by: 1 1 EN + GL − 2M F + = R1 R2 EG − F 2 and the mean curvature is given by: 1 1 1 1 EN + GL − 2M F H≡ + . = 2 R1 R2 2 EG − F 2

(B.12)

(B.13)

Since the sign of ¯ 1n is not yet deﬁned, the sign of H is not determined. B.2.2 Mean Curvature of an Axially Symmetric Surface If the surface S is axially symmetric, its equation is given by: S(θ, z) ≡ r(θ, z) = r(z)¯ 1r + z ¯1z and leads to1 : 1

() =

d() . dz

(B.14)

240

B Geometry Complements

⎧ ⎪ ⎨E = F = ⎪ ⎩ G=

∂r ∂r ∂θ . ∂θ ∂r ∂r ∂θ . ∂z ∂r ∂r ∂z . ∂z

= r2 . =0 2 =1+r

(B.15)

According to (B.5): 1z + ¯ 1r −r ¯ ¯ 1n = √ 2 1+r that, combined with (B.4), leads to: ⎧ r ⎪ ⎨ L = − √1+r2 M =0 . ⎪ ⎩ N = √ r 1+r 2

(B.16)

(B.17)

The mean curvature of an axially symmetric surface is consequently written as: 1 r 1 H= − (B.18) 2 (1 + r2 )3/2 r(1 + r2 )1/2

B.3 Catenary Curve In the case of ∆p = 0, (8.13) can be rewritten as: dr 2 ) 1 + ( dz d2 r . = dz 2 r

The solution to this equation looks like: z−B r(z) = A cosh , C z−B A dr = sinh , dz C C z−B A d2 r = 2 cosh , dz 2 C C

(B.19)

(B.20) (B.21) (B.22)

and leads to the condition A = C. A and B can be determined with the following boundary conditions: r(b) = a, 1 , r (b) = tan θ where a, b, and θ are represented in Fig. B.2.

(B.23) (B.24)

B.3 Catenary Curve

241

x 10−3 2

z [m]

1.5

1 Analytical solution Numerical solution

0.5

0

0

0.5

1

1.5 r [m]

2

2.5

3 x 10−3

Fig. B.2. Boundary conditions: a, b and θ

These equations are transformed into: b−B a = A cosh A b−B 1 = sinh tan θ A and (B.25) can be written as: b−B a = A 1 + sinh2 A 1 = A 1+ tan2 θ A =± sin θ ⇒ A = ±a sin θ.

(B.25) (B.26)

(B.27) (B.28) (B.29) (B.30)

By using this result in (B.25): 1 B = b ∓ a sin θa cosh sin θ

(B.31)

because a cosh is a pair function. As a is a positive radius and 0 < θ < π: A = a sin θ, B = b − a sin θ a cosh

1 sin θ

(B.32)

.

(B.33)

C Comparison Between Both Approaches

This appendix shows that the expression under brackets in (9.47), hereafter noted B in (C.1), is equal to −(cos θ1 + cos α)/ tan φ, which leads to (9.48). This step is used in the demonstration of equivalence of the Laplace equation based method and the energy minimization method used to compute capillary forces.

1+

1

sin α cos α+2 sin α cos θ1 −π+α+θ1 −sin θ1 cos θ1 (cos θ1 +cos α)2

tan φ

.f

=

≡n

=−n1

=−n2

2 . (π − α − θ ) sin φ − sin α cos θ sin φ + sin θ cos θ sin φ −(cos θ cos φ + cos θ )(cos θ + cos α) / 1 1 1 1 1 2 1 . ... cos φ(cos θ1 + cos α)2 + (sin α cos α + 2 sin α cos θ1 −π + α + θ1 − sin θ1 cos θ1 ) sin φ

≡n1

cos φ(cos θ1 + cos α)2 .f cos φ(cos θ1 + cos α)2 + (sin α cos α + 2 sin α cos θ1 − π + α + θ1 − sin θ1 cos θ1 ) sin φ cos φ(cos θ1 + cos α) = ... sin φ

B=

Using (9.27), µ can be replaced, leading to:

≡f

The expression under brackets (let us note it B) can be reduced to a common denominator: 1 cos θ1 cos θ2 π − α − θ1 − sin α cos θ1 + sin θ1 cos θ1 − + B= (cos θ1 + cos α)(1 + µ tan φ) 1 + µ tan φ tan φ sin φ (π − α − θ1 ) sin φ − sin α cos θ1 sin φ + sin θ1 cos θ1 sin φ − (cos θ1 cos φ + cos θ2 )(cos θ1 + cos α) 1 = . 1 + µ tan φ (cos α + cos θ1 ) sin φ

≡B

cos θ1 cos θ2 D . π − α − θ1 − sin α cos θ1 + sin θ1 cos θ1 1 cos θ1 cos θ2 / dW = + + − ( + ) . 2Lγ dD tan φ sin φ D+h (cos θ1 + cos α)(1 + µ tan φ) 1 + µ tan φ tan φ sin φ

(C.2)

(C.1)

244 C Comparison Between Both Approaches

−(n1 +n2 )

=−1

(C.3)

Adding and subtracting sin α, the latter equation can be written as follows: dW cos θ1 cos θ2 D cos φ(cos θ1 + cos α) = + − − sin α + sin α 2Lγ dD tan φ sin φ D+h sin φ (cos θ1 cos φ + cos θ2 )(D + h) − D cos φ(cos θ1 + cos α) − sin α sin φ(D + h) = + sin α (D + h) sin φ = ... h cos φ(cos θ1 + cos θ2 cos φ − sin θ2 sin φ) = + sin α (D + h) sin φ h cos θ1 + cos α = sin α + . D+h tan φ

and the surface energy derivative given by (9.47) can be rewritten into: dW cos θ1 cos θ2 D cos φ(cos θ1 + cos α) = + − . 2Lγ dD tan φ sin φ D+h sin φ

(C.6)

(C.5)

Now, by replacing α by α = θ2 + φ and simplifying the latter expression, it can be shown that the expression under brackets in (C.3) is equal to −1. Consequently, B is given by: cos φ(cos θ1 + cos α) (C.4) B=− sin φ

. (π − α − θ + sin θ cos θ ) sin φ − sin α cos θ sin φ − (cos θ cos φ + cos θ )(cos θ + cos α) / 1 1 1 1 1 2 1 ... . (−π + α + θ1 − sin θ1 cos θ1 ) sin φ + cos φ(cos θ1 + cos α)2 + (sin α cos α + 2 sin α cos θ1 ) sin φ

n1 +n2

Let us rearrange the bracket of the latter equation in order to put n1 and n2 forward: cos φ(cos θ1 + cos α) ... B= sin φ

C Comparison Between Both Approaches 245

246

C Comparison Between Both Approaches

The latter equation is equivalent to (9.37), which demonstrates the equivalence between the Laplace equation based and the energetic force formulations.

D Symbols

Table D.1. Greek symbols Symbol α β δ δe ∆p φ γ γLV γSL γSV µ ρ ρ ρ Σ θ θA θR θ1 θ2

Description Angular beam deﬂection Filling angle Beam deﬂection Beam deﬂection at calibration Pressure diﬀerence across the LV interface = pin − pout Misalignment Residual adhesion ratio Surface tension of a liquid Energy of the interface Liquid–Vapor Energy of the interface Solid–Liquid Energy of the interface Solid–Vapor Dynamic viscosity Principal curvature radius of the meniscus (usually < 0) Principal curvature radius of the meniscus (usually > 0) Density Lateral area of the liquid bridge Contact angle Advancing contact angle Receding contact angle Contact angle at the component side Contact angle at the gripper side

Unit deg deg m m Pa m, deg − N m−1 J m−2 J m−2 J m−2 Pa s m m kg m−3 m2 rad rad rad rad rad

248

D Symbols

Table D.2. Latin symbols Symbol a b Bo C Ca d D dpinc dpstart E F FL FT g G k K h

H I L LC m0 M N p p1 , p2 pin pout r1 r2 rneck R

S U ULV v V

Description Acceleration Beam width Bond number Capacity of the SL interface in electrowetting Capillary number (nondimensional) Characteristic size of the meniscus in nondimensional analysis Diameter of a droplet posed on the substrate gap in Chap. 9 Pressure increment in θ1 search Initial pressure diﬀerence in θ1 search Young’s modulus Diﬀerential operator to compute H Diﬀerential operator to compute H Laplace term of the capillary force Interfacial tension term of the capillary force Earth gravity Diﬀerential operator to compute H Stiﬀness of the cantilevered beam Interaction constant depending on materials (VDW) Beam thickness Height of a droplet posed on the substrate Immersion height Capillary rise Mean curvature Inertia moment of the beam section Length of the cantilevered beam Diﬀerential operator to compute H Capillary length Standardized mass(es) used for calibration Diﬀerential operator to compute H Diﬀerential operator to compute H Gripper parameter (radius or angular aperture) Permanent dipoles (VDW) Pressure in the liquid phase Pressure in the vapor phase Radius of the liquid bridge at the component side Radius of the liquid bridge at the gripper side Radius of the liquid bridge at the neck height Radius of a droplet posed on the substrate Radius of spherical grippers Molar gas constant (8.314) Area of a spherical cap Total interfacial energy of the droplets (S–V, L–V, S–L) Partial interfacial energy of the droplets (L–V) Molar volume Volume of liquid, usually between 0.1 and 1 µL

Unit m s−2 m − F − m m Pa Pa GPa

N N 9.81 m s−2 N m−1 J m6 m m m m m−1 m4 m m kg

m or deg Pa Pa m m m m m J K−1 mol−1 m2 J J m3 mol−1 m3

D Symbols Table D.3. Latin symbols Symbol Description W Component weight Total interfacial energy of the meniscus (S–V, L–V, S–L) Partial interfacial energy of the meniscus (L–V) WLV z Separation distance between the gripper and the object Approaching contact distance before the droplet zA hung to the gripper contacts the object Distance at which the meniscus breaks z∗ and is replaced by two droplets Table D.4. Abbreviations Abb. AFM CSEM DIN DMT EAP EDM JKR LAB LIGA LPM LRP MEL MEMS MOC RH SEM SFA SMA SMD VDW

Description Atomic force microscope Swiss Centre for Electronics and Microtechnology German norms Derjagin–Muller–Toporov Electroactive polymer Electrodischarge machining Johnson–Kendall–Roberts model Automation Laboratory of Besan¸con Lithographie-Galvanoformung-Abformung Microtechnology production laboratory (EPFL) Robotics laboratory of Paris (Paris VI) Mechanical engineering laboratory Micro Electro Mechanical System Microrobot on chip Relative humidity Scanning electron microscope Surface force apparatus Shape memory alloy Surface mount device van der Waals

Unit N J J m m m

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Index

Accuracy, 152 Adhesion, XIV, 42 Advancing contact angle, 161 AFM, XI, 19, 20, 29, 33 Anova, 189, 194 Approximations Arc, Circle, Toroidal, 51 Parabolic, 51 Assembly Combined part manufacturing and assembly, 6 Design for microassembly, 4 Microassembly literature, 9 Parallel assembly, 8 Performances of the assembly machines, 103 Product external/internal assembly functions, 6 Requirements, 3 Self-assembly, 7 Stochastic assembly, 7 Taxonomy, 4 Assumptions of the static simulation, 65 Bernoulli, 22 Bond number, 51, 76, 105 Boundary conditions Equilibrium, 69 Buckingham theorem, 103 CAD model, 148 Calibration, 151

Capillary Gripper, 20, 27, 47 Length, 51, 66 Capillary force, 45 Between a sphere and a plane, 55 Between two parallel plates, 52 Between two spheres, 56 Laplace term, 46 Surface tension term, 46 Case study, 83, 127, 189, 199 Casimir, 11 Cassie, 97, 137 Catenary curve, 164, 240 CCD, 148 Circle approximation See Approximations, 51 Components, 159 Condensation, 101 Contact Distance of, 185 Line, 42, 43, 100 Contact angle, 43, 46, 112, 168, 174 Advancing contact angle, 161, 168 Dynamic contact angle, 100 Hysteresis, 99 Measurement, 154 Receding contact angle, 161, 168 Coulomb, 10, 11 Cryogenic gripper, 21, 27 Curvature See Mean curvature, 45 Curvature radius, 68

262

Index

Damping Eﬀect, 101 Time, 106, 108 Debye, 11 DFµA, 4 Dimensional Analysis, 103, 194 Matrix, 103 DIN, 29 Dispensing device, 146 DMT, 12 Drop generation, 146 Du Nou¨ y, 155, 191 Dynamical release, 177

Push-pull, 19 Ultrasonic levitation, 23 Vacuum, 15 GS-St-#, 158

EAP, electroactive polymer, 21 EDM, 5 Energetic Capillary force approximation, 51 Energy, 51 Interfacial energy W , 51, 52

JKR, 12, 17

Forces Analytical approximations for capillary forces, 227 Capillary force, 45, 47 Classiﬁcation, 10 Coulomb, 10 Force-distance curve, 170 Laplace force, 46 Measurement, 145 Shear, 186 Tension force, 46, 171 Froude, 105 Gap, 46, 170 GC-St-#, 158 Gripper Aerodynamic levitation, 22 Air cushion levitation, 22 Available grippers, 158 Capillary, 20, 27, 47 Cryogenic, 21, 27 Electrostatic, 17 Form closure, 15 Friction based, 13 Magnetic, 16 Optical, 24 Parameter, 113, 171

Hagen-Poiseuille, 106 Heterogeneities, 97 Hysteresis, 99 Impurities, 97 Interfacial energy, 52 Interfacial tension force See Forces, 46 Israelachvili, 20, 55, 81, 93, 99, 171, 212

Keesom, 11 Kelvin Equation, 99 Radius, 99 Lack of ﬁt, 194 Laplace, 137 Equation, 43, 45, 61, 65 Force, 46, 171 Lifshitz, 10, 11 LIGA, 13 Limits, 103, 108 Linearity, 151 Error, 151 London, 10 Lucas-Washburn, 103, 138 Mean curvature, 45, 238 MEL, 5 MEMS, 11, 42, 101 Meniscus proﬁle, 163 Microworld, IX Mikromasch, 20 Misalignment, 152 Non dimensional numbers, 103 Numerical simulation, 65 Orientation (Relative), 174 Parabolic approximation, 51, 61 Picking strategies, 211 Polder, 11

Index Protocol See Test bed, 155 PTFE, 176 R47V50, 161 Receding contact angle, 161 Relative humidity, 18 Release Dynamical, 177 Releasing strategies, 213 Repartition, 233 Requirements Microassembly, 3 Test bed, 143 Reynolds, 105 Roughness Inﬂuence on wetting properties, 98 Rupture, 111, 185 Rupture gap, 117 SEM, 15 SFA, 101 Shape memory alloy, 29 Shear force, 186 Shooting method, 70 Si-#, 159 Simulation Algorithms, 127 Assumptions, 65 Limits of, 103 Numerical, 65 Static, 65 Validation of, 163 Spreading, 100

263

St-#, 159 Stiction, 42 Surface Energy, 51 Impurities, 97 Mount Device, 35 Tension, 41, 172 Symbols, 247 Tension force, 171 Test bed, 143 Accuracy, 152 Available components, 159 Available grippers, 158 Calibration, 151 Linearity, 151 Misalignment, 152 Modus operandi, 155 Tilt, 174 Toroidal approximation See Approximations, 51 Triple line See Contact line, 42 van der Waals, 10, 11, 25, 27, 42, 176 Vision, 148 Volume repartition, 111, 115, 233 Weber, 105 Wenzel, 137 Wilhelmy, 155 Young-Dupr´e, 42, 137 Zisman model, 99