# Papers of Ken Brakke

K. Brakke, Some new values of Sylvester's function for n noncollinear points, J. Undergrad. Math. 4 (1972) 11-14.
K. Brakke, The Motion of a Surface by its Mean Curvature, Princeton University Press, 1978. The expanded book version of my Ph.D. thesis. Now out of print, so this is a scan of the book in PDF format (only 8MB!). And a LaTeX version, with a table of notation and some misprints corrected.
J. Mantock, K. Fukunage, and K. Brakke, Systematic feature extraction, IEEE Transactions on Pattern Analysis and Machine Intelligence PAMI-4 (1982) 291-297. PDF 1.1MB
Voronoi tessellation manuscripts. Five unpublished papers on simulating Voronoi tessellations that I did in the mid-1980's.
K. Brakke, Minimal cones on hypercubes, J. Geom. Anal. vol. 1 (1991) 329-338.
PDF file, 130KB.

Abstract: It is shown that in dimension greater than 4, the minimal area hypersurface separating the faces of a hypercube is the cone over the edges of the hypercube. This constrasts with the cases of two and three dimensions, where the cone is not minimal. For example, a soap film on a cubical frame has a small rounded square in the center. In dimensions over 6, the cone is minimal even if the area separating opposite faces is given zero weight. The proof uses the maximal flow problem that is dual to the minimal surface problem.

K. Brakke, The Surface Evolver, Experimental Mathematics vol. 1 no. 2 (1992), 141-165.
PDF file, 400KB.

Abstract: The Surface Evolver is a computer program that minimizes the energy of a surface subject to constraints. The surface is represented as a simplicial complex. The energy can include surface tension, gravity, and other forms. Constraints can be geometrical constraints on vertex positions or constraints on integrated quantities such as body volumes. The minimization is done by evolving the surface down the energy gradient. This paper describes the mathematical model used and the operations available to interactively modify the surface.

K. Brakke, The Opaque Cube Problem, Am. Math. Monthly vol. 99 (Nov. 1992), 866-871.
PDF file, 240KB.

Abstract: It is a classic puzzle to find the shortest set of curves that intersect all straight lines through a square, and the conjectured solution is still unproven. This paper asks the analogous question for a cube, and comes up with the best known solution.

K. Brakke, Minimal Surfaces, Corners, and Wires, J. Geom. Anal. vol. 2 no. 1 (1992), 11-36.
PDF file, 940KB.

Abstract: Weierstrass representations are given for minimal surfaces that have free boundaries on two planes that meet at an arbitrary dihedral angle. The contact angles of a surface on the planes may be different. These surfaces illustrate the behavior of soapfilms in convex and nonconvex corners. They can also be used to show how a boundary wire can penetrate a soapfilm with a free end, as in the overhand knot surface. They should also cast light on the behavior of capillary surfaces.

K. Brakke, Soap films and covering spaces, J. Geom. Anal. vol. 5 no. 4 (1995) 445-514.
PDF file, 3MB.

Abstract: A new mathematical model of soap films is proposed, called the "covering space model." The two sides of a film are modelled as currents on different sheets of a covering space branching along the film boundary. Hence a film may be seen as the minimal cut separating one sheet of the covering space from the others. The film is thus the oriented boundary of one sheet, which represents the exterior of the film. As oriented boundaries, films may be calibrated with differential forms on the covering space, a version of the min-cut, max-flow duality of network theory. This model applies to unoriented films, films with singularities, films touching only part of a knotted curve, films that deformation retract to their boundaries, and other examples that have proved troublesome for previous soap film models.

K. Brakke, Numerical Solution of Soap Film Dual Problems, Experimental Mathematics vol. 4 no. 4 (1995) 269-287.
PDF file, 1MB (and cover illustration)

Abstract: The soap film problem is to minimize area, and its dual is to maximize the flux of a divergenceless bounded vectorfield. This paper discretizes the continuous problem and solves it numerically. This gives upper and lower bounds on the area of the globally minimizing film. In favorable cases, the method can be used to discover previously unknown films. No initial assumptions about the topology of the film are needed. The paired calibration or covering space model of soap films is used to enable representation of films with singularities.

K. Brakke, R. Phelan, and D. Weaire, Computation of equilibrium foam structure using the Surface Evolver, Experimental Mathematics 4 (1995) 181-192.
PDF file, 5.5MB. (and cover illustration)

Abstract: The Surface Evolver has been used to minimise the surface area of various ordered structures for monodisperse foam. Additional features have enabled its application to foams of arbitrary liquid fraction. Early results for the case of dry foam (negligible liquid fraction) produced a structure haveing lower surface area, or energy, than Kelvin's 1887 minimal tetrakaidecahedron. The calculations reported here show that this remains the case when the liquid fraction is finite, up to about 11%, at which point an f.c.c arrangement of the cells becomes preferable.

K. Brakke and T. Singler, Computer simulation of solder bridging phenomena, Transactions of the ASME 118 (1996) 122-126.

Abstract: Solder bridging is investigated under the assumption that liquid solder bridges are equilibrium capillary surfaces and that the principal factor that determines whether a bridge will freeze to form a permanent short is its configurational stability. A computational paramemtric bridge stability study is conducted to determine the response of bridging to the system volume, the distance between pads, the contact angle between the liquid metal ant resist surface and the relevant physiochemical properties of the liquid metal.

K. Brakke, The Surface Evolver and the stability of liquid surfaces , Phil. Trans. R. Soc. A vol. 354 (1996) 2143-2157.
PDF file, 1MB

Abstract: The Surface Evolver is an interactive program for studying the shapes of liquid surfaces. Recently added features permit the calculation of the Hessian matrix of second derivatives of the energy. The Hessian can be used for fast convergence to an equilibrium, and eigenvalue analysis of the stability of that equilibrium. This paper describes the use of the Hessian by the Surface Evolver, presents some sample stability analyses, and gives some numerical results on the accuracy and convergence of the methods. It is also shown how one can evolve unstable surfaces.

G. Francis, J. M. Sullivan, R. Kusner, K. Brakke, C. Hartman, and G. Chapell, The minimax sphere eversion, in Mathematics and Visualization, ed. K. Polthier and H. Hege, Springer-Verlag, Berlin, 1997, 3-20.
PDF file, 2MB

Abstract: We consider an eversion of a sphere driven by a gradient flow for elastic bending energy. We start with a halfway model which is an unstable Willmore sphere with 4-fold orientation-reversing rotational symmetry. The regular homotopy is automatically generated by flowing down the gradient of the energy from the halfway model to a round sphere, using the Surface Evolver. This flow is not yet fully understood; however, our numerical simulations give evidence that the resulting eversion is isotopic to one of Morin's classical sphere eversions. These simulations were presented as real-time interactive animations in the CAVE automatic virtual environment at Supercomputing'95, as part of an experiment in distributed, parallel computing and broad-band, asynchronous networking.
Video available.

K. Brakke and J. Sullivan, Using symmetry features of the Surface Evolver to study foams, in Mathematics and Visualization, Springer Verlag, Berlin, 1997, ed. Konrad Polthier and Hans-Christian Hege, 95-118.
PDF file, 1.3MB

Abstract: This paper describes the use of various symmetry features, including periodic boundary conditions, mirror boundaries, and rotational symmetry, in the Evolver. As a test case, we use these features to study foams, in particular the equal-volume foams of Kelvin and Weaire-Phelan. To compute the shape and energy of one of these compound surfaces, it is most efficient to work with only the smallest possible fundamental domain.

F. Baginski and K. Brakke, Modeling ascent configurations of strained high-altitude balloons, AIAA Journal 36 (1998) 1901-1910.
PDF file, 2MB.

Abstract: We consider the problem of estimating stresses in the ascent shape of an elastic high-altitude scientific balloon. The balloon envelope consists of a number of long, flat, tapered sheets of polyethylene called gores that are sealed edge-to-edge to form a complete shape. Because the film is so thin, it has zero bending stiffness and cannot support compressions. In particular, the balloon film forms internal folds of excess material when the volume is not sufficiently large. Because of these factors, a standard finite element approach will have difficulty computing partially inflated balloon shapes. In our approach, we develop a variational principle for computing strained balloon shapes that incorporates regions of folded material as a part of the geometric model. We can apply our model to fully inflated or partially inflated configurations. The equilibrium shape is the solution of minimum energy satisfying a given volume constraint. We apply our model to a design shape representative of those used in scientific ballooning and compute a family of ascent configurations with regions of external contact for a volume as low as 22% of its float value.

K. Brakke and F. Morgan, Instability of the wet X soap film, J. Geom. Anal. 8, no. 5 (1998), 749-768.
PDF file, 680KB.

Abstract: For idealized, infinitely thin ("dry") soap films, an X is unstable, while for very thick ("wet") soap films it is minimizing. We show that for soap films of relatively small but positive wetness, the X is unstable. Full stability diagrams for the constant liquid fraction case and the constant pressure case are generated. Analogous questions about other singularities remain controversial.

K. Brakke and F. Morgan, Instabilities of cylindrical bubble clusters, Eur. Phys. J. E 9 (2002) 453-460.
PDF file, 650 KB

Abstract: Small bubbles in an experimental two-dimensional foam between glass plates regularly undergo a three-dimensional instability as the small bubbles shrink under diffusion or equivalently as the plate separation increases, and end up on one of the plates. The most recent experiments of Cox, Weaire, and Vaz are accompanied by Surface Evolver computer simulations and rough theoretical calculations. We show how a recent second variation formula may be used to perform exact theoretical calculations for infinitesimal perturbations for such a system, and verify results with Surface Evolver simulations.

K. Brakke, Instability of the wet cube cone soap film, Colloids and Surfaces A: Physiochem Eng. Aspects 263 (2004) 4-10.
PDF file, 389 KB.

Abstract: A "dry" conical soap film on a cubical frame is well known not to be stable. Recent experimental evidence seems to indicate that adding liquid to form "Plateau borders" stabilizes the conical film, perhaps to arbitrarily low liquid volumes. This paper presents numerical simulation evidence that the wet cone is unstable for low enough liquid volume, with the critical volume fraction being about 0.000274.

F. Baginski, K. Brakke, and W. Schur, Stability, clefting, and other issues related to undesirable equilibria in large pumpkin balloons, 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference; Austin, TX; USA; 18-21 Apr. 2005. pp. 1-13.
PDF file, 430 KB.

Abstract: NASA's effort to develop a large payload, high altitude, long duration balloon, the Ultra Long Duration Balloon, focuses on a pumpkin shape super-pressure design. It has been observed that a pumpkin balloon may be unable to pressurize into the desired cyclically symmetric equilibrium configuration, settling into a distorted, undesired state instead. Hoop stress considerations in the pumpkin design leads to choosing the lowest possible bulge radius, while robust deployment is favored by a large bulge radius . Some qualitative understanding of design aspects on undesired equilibria in pumpkin balloons has been obtained via small-scale balloon testing. Poorly deploying balloons have clefts, but most gores away from the cleft deploy uniformly. Mechanical locking may be a contributing factor in the formation of such undesired equilibria. Long term success of the pumpkin balloon for NASA requires a thorough understanding of the phenomenon of multiple stable equilibria. This paper uses the notion of stability to cl assify balloon designs. When we applied our model to a balloon based on the NASA Phase IV-A pumpkin design, we found the fully inated/fully deployed strained equilibrium float configuration to be unstable. To explore the sensitivity of this particular design and to demonstrate our general approach, we carry out a number of parametric studies that are variations on the Phase IV-A design. In this paper, we will focus on analytical studies, but we also compare our results with experimental and flight data wh enever possible. We will discuss the connection between stability and the generic deployment problem.

D. Weaire, S. Cox, and K. Brakke, Liquid Foams, book chapter in Cellular Ceramics, ed. P. Colombo and M. Scheffler, Wiley-VCH, Weinheim, 2005.
PDF file, 460 KB

Abstract: The elegant structure of a liquid foam and its constituent parts have fascinated scientists for centuries. A combination of experiments, theory and simulations has elucidated most of its static and quasi-static properties. However, this is only part of a wider subject: dynamic effects remain as a considerable challenge, particularly for wet foams.

M. Anderson, C. Egger, J. Casci, G. Tiddy, and K. Brakke, A new minimal surface and the structure of mesoporous silicas, Angew. Chem. Int. Ed. 44 (2005) 2-6. With supporting material.
PDF files, 670 KB and 315 KB

Abstract: We report our studies of the structure of the surfactant-templated, cubic, mesoporous silica superstructure SBA-1 and provide a formulation in terms of curvature that has important repercussions for both surfactant structures and the mechanism of formation of inorganic replicas. We establish that the crucial interface that determines the inorganic structure is between the silica and water adsorbed at the micelle surface, not between silica and surfactant, thus challenging the present synth esis me chanisms. We adopt a general protocol for understanding the surface curvature and energy which could be applied widely to the growth of inorganic structures in biology, including nonperiodic and disordered structures.

F. Baginski, K. Brakke, and W. Schur, Stability of cyclically symmetric strained pumpkin balloon configurations and the formation of undesired equilibria, Journal of Aircraft 43, no. 5, (2006) 1414-1423.
PDF file, 3MB

Abstract: NASA'S effort to develop a large payload, high altitude, long-duration balloon, the ultralong duration balloon, focuses on a pumpkin shape superpressure design. It has been observed that a pumpkin balloon may be unable to pressurize into the desired cyclically symmetric equilibrium configuration, settling into a distorted, undesired state instead. Hoop stress considerations in the pumpkin design lead to choosing the lowest possible bulge radius, whereas robust deployment is favored by a large bul ge radius. Mechanical locking may be a contributing factor in the formation of undesired equilibria. Long term success of the pumpkin balloons for NASA requires a thorough understanding of the phenomenon of multiple stable equilibria. This paper uses the notion of stability to classify balloon designs. When we applied our finite element model to a balloon based on the NASA Phase IV-A pumpkin design, we found the fully inflated/fully deployed strained equilibrium float configuration was unstable. To demonst rate our approach for exploring the stability of constant bulge radius designs and their sensitivity to parameter changes we carry out a number of parametric studies. We focus on analytical studies, but we also compare our resuts with flight data whenever possible.

F. Baginski, K. Brakke, and W. Schur, Unstable, cyclically symmetric and stable, asymmetric pumpkin-balloon configurations, Journal of Aircraft 44, no. 3, (2006) 764-773.
PDF file, 4MB

Abstract: By design, a pumpkin balloon is intended to assume a cyclically symmetric "pumpkin-like" shape once it reaches float altitude and is fully inflated. Recent work by the authors showed that under certain circumstances, a strained cyclically symmetric pumpckin balloon configuration can be unstable. This means the balloon must assume an alternate non-cyclically symmetric stable equilibrium shape. Julian Nott's round-the-world balloon Endeavoru was on of the first pumpkin-type balloons to encounter this instability. In this paper, we will explore the phenomena of unstable cyclically symmetric and stable asymmetric balloon configurations.

F. Baginski, K. Brakke, and W. Schur, Cleft formation in pumpkin balloons, Advances in Space Research 37 no. 11 (2006) 2070-2081.
PDF file, 3MB

Abstract: NASA's development of a large payload, high altitude, long duration balloon, the Ultra Long Duration Balloon, centers on a pumpkin shape super-pressure design. Under certain circumstances, it has been observed that a pumpkin balloon may be unable to pressurize into the desired cyclically symmetric equilibrium configuration, settling into a distorted, undesired state instead. In this paper, we will use th concept of stability to classify equilibrium shapes of fully pressurized/fully deployed strained ball oons.

G. C. Shearman, B. J. Khoo, M. Motherwell, K. A. Brakke, O. Ces, C. E. Conn, J. M. Seddon, and R. H. Templer, Calculations of and evidence for chain packing stress in inverse lyotropic bicontinuous cubic phases, Langmuir 23 (2007) 7276-7285.
PDF file, 470 KB

Abstract: Inverse bicontinuous cubic lyotropic phases are a complex solution to the dilemma faced by all self-assembled water-amphiphile systems: how to satisfy the incompatible requirements for uniform interfacial curvature and uniform molecular packing. The solution reached in this case is for the water-amphiphile interfaces to deform hyperbolically onto triply periodic minimal surfaces. We have previously suggested that although the molecular packing in these structures is rather uniform the relative phase behavior of the gyroid, double diamond, and primitive inverse bicontinuous cubic phases can be understood in terms of subtle differences in packing frustration. In this work, we have calculated the packing frustration for these cubics under the constraint that their interfaces have constant mean curvature. We find that the relative packing stress does indeed differ between phases. The gyroid cubic has the least packing stress, and at low water volume fraction, the primitive cubic has the greatest packing stress. However, at very high water volume fraction, the double diamond cubic becomes the structure with the greatest packing stress. We have tested the model in two ways. For a system with a double diamond cubic phase in excess water, the addition of a hydrophobe may release packing frustration and preferentially stabilize the primitive cubic, since this has previously been shown to have lower curvature elastic energy. We have confirmed this prediction by adding the long chain alkane tricosane to 1-monoolein in excess water. The model also predicts that if one were able to hydrate the double diamond cubic to high water volume fractions, one should destabilize the phase with respect to the primitive cubic. We have found that such highly swollen metastable bicontinuous cubic phases can be formed within onion vesicles. Data from monoelaidin in excess water display a well-defined transition, with the primitive cubic appearing above a water volume fraction of 0.75. Both of these results lend support to the proposition that differences in the packing frustration between inverse bicontinuous cubic phases play a pivotal role in their relative phase stability.

D-Y. Shin, K. A. Brakke, Theoretical analysis of jetting and ink filling processes for TFT LCD colour filters, Proceedings of the 10th Asian Symposium on Information Display, August, 2007, Singapore.
PDF file, 217 KB

Abstract: The fabrication of TFT LCD colour filters with the piezo Drop-On-Demand (DOD) inkjet printing technology has gained attention from industries. However, this technology differs from previous processes such as spin and slit coating technologies in terms of the degree of complexity. Different from spin and slit coating processes, the piezo DOD inkjet printing technology has the capability to selectively deposit ink droplets on the positions, which greatly saves the waste of materials in producing TFT LCD colour filters. This feature, however, draws two engineering difficulties. First, the ink droplet volume should be carefully controlled to avoid the total ink volume variation among subpixels, which, otherwise, could cause visible swathe marks. Second, ink droplets must be confined without the introduction of unfilled regions in a subpixel and spilling over into the adjacent subpixels. In this study, two fundamental theoretical analyses are performed to investigate one possible cause of visible swathe marks and suggest a concise way to derive the optimum surface conditions which eventually confine ink in a subpixel.

D-Y. Shin, K. A. Brakke, Theoretical Investigation of Jetting and Wetting Phenomena for the Fabrication of TFT LCD Color Filters, August, 2007, Daegu, South Korea, 376-379.
PDF file, 217 KB

Abstract: Although years of trials for the fabrication of TFT LCD color filters with the piezo Drop-On-Demand (DOD) inkjet printing technology have been made, the underlying physics of jetting and wetting has not been fully understood. In this study, the key engineering issues, jetting and wetting, are investigated with mathematical models.

J. Berthier1, K. Brakke, F. Grossi, L. Sanchez, L. Di Cioccio Silicon Die Self-alignment on a Wafer: Stable and Unstable Modes, Sensors & Transducers Journal, 115(4), 135-150 (2010).
PDF file, 1.3 MB

Abstract: 3D integration is the key to advanced microelectronic systems. Die-to-wafer assembly is a necessary step to reach full integration. Self-assembly methods are promising due to their parallel aspect which overcomes the main difficulties of the current techniques. The aim of this work is the understanding of the mechanisms of self-alignment with an evaporating droplet technique and the investigation the stable and unstable modes. Using the Surface Evolver software, we analyze the causes for misalignments of the system and their evolution.

J. Berthier1, K. Brakke, F. Grossi, L. Sanchez, L. Di Cioccio Self-alignment of silicon chips on wafers: a capillary approach, Journal of Applied Physics 1-8, 054905 (2010). {br} PDF file, 1.2 MB

Abstract:As the limits of Moores law are approached, three-dimensional integration appears as the key to advanced microelectronic systems. Die-to-wafer assembly appears to be an unavoidable step to reach full integration. While robotic methods experience difficulties to accommodate fabrication speed and alignment accuracy, self-assembly methods are promising due to their parallel aspect, which overcomes the main difficulties of current techniques. The aim of this work is the understanding of the mechanisms of self-alignment with an evaporating droplet technique. Stable and unstable modes are examined. Causes for misalignments of chips on wafers and their evolution are investigated with the help of the SURFACE EVOLVER numerical software. Precautions for suitable alignment are proposed.

R. Gabrielli, A. Meagher, D. Weaire, S. Hutzler, K. A. Brakke, An experimental realisation of the Weaire-Phelan structure in monodisperse liquid foam, Phil. Mag. Letters, 92(1), 1-6 (2012).
PDF file, 1.7 MB

Abstract: The WeairePhelan (WP) structure is the lowest energy structure known of an ideal monodisperse foam in the dry limit. To date, it has not been realized in the laboratory. Instead Lord Kelvins 1887 structure, which it supplanted in 1994, has repeatedly been found in attempts to produce an ordered structure. This paradox is attributable to the flat walls of the containers used, with which the Kelvin structure is more compatible. Accordingly, we have fabricated a patterned mould whose faceted walls conform to the WP geometry, and thereby succeeded in inducing the formation of perfect crystals of the WP structure. Foam samples consisted of approximately 1500 bubbles. Vibrations favoured crystallization.

Rodrigo A. Rezende, Frederico D.A.S. Pereira, Vladimir Kasyanov, Aleksandr Ovsianikov, Jan Torgensen, Peter Gruber, Jurgen Stampfl, Ken Brakke, Jlia A. Nogueira, Vladimir Mironov, Jorge V.L. da Silva Design, physical prototyping and initial characterisation of `lockyballs', Virtual and Physical Prototyping 7(4) 287-301 (2012)
PDF file, 1.1 MB

Abstract: irected tissue self-assembly or bottom-up modular approach in tissue biofabrication is an attractive and potentially superior alternative to a classic top-down solid scaffold-based approach in tissue engineering. For example, rapidly emerging organ printing technology sing self-assembling tissue spheroids as building blocks is enabling computer-aided robotic ioprinting of three-dimensional (3D) tissue constructs. However, achieving proper aterial properties while maintaining desirable geometry and shape of 3D bioprinted issue engineered constructs using directed tissue self-assembly, is still a challenge. roponents of directed tissue self-assembly see the solution of this problem in developing ethods of accelerated tissue maturation and/or using sacrificial temporal supporting of emovable hydrogels. In the meantime, there is a growing consensus that a third strategy ased on the integration of a directed tissue self-assembly approach with a conventional olid scaffold-based approach could be a potential optimal solution. We hypothesise that issue spheroids with velcro-like interlockable solid microscaffolds or simply lockyballs ould enable the rapid in vivo biofabrication of 3D tissue constructs at desirable material roperties and high initial cell density. Recently, biocompatible and biodegradable photo- ensitive biomaterials could be fabricated at nanoscale resolution using two-photon olymerisation (2PP), a development rendering this technique with high potential to abricate velcro-like interlockable microscaffolds. Here we report design studies, physical rototyping using 2PP and initial functional characterisation of interlockable solid icroscaffolds or so-called lockyballs. 2PP was used as a novel enabling platform echnology for rapid bottom-up modular tissue biofabrication of interlockable constructs. he principle of lockable tissue spheroids fabricated using the described lockyballs as solid icroscaffolds is characterised by attractive new functionalities such as lockability and unable material properties of the engineered constructs. It is reasonable to predict that these building blocks create the basis for a development of a clinical in vivo rapid biofabrication approach and form part of recent promising emerging bioprinting technologies.

J. Berthier, S. Mermoz, K. Brakke, L. Sanchez, C. Fretigny, and L. Di Cioccio Capillary self-alignment of polygonal chips: A generalization for the shift-restoring force, Microfluidics and Nanofluidics 14(5), 845-858 (2013).
PDF file, 2.3 MB

Abstract: Capillary-driven self-alignment using droplets is currently extensively investigated for self-assembly and microassembly technology. In this technique, surface tension forces associated to capillary pinning create restoring forces and torques that tend to bring the moving part into alignment. So far, most studies have addressed the problem of square chip alignment on a dedicated patch of a wafer, aiming to achieve 3D microelectronics. In this work, we investigate the shift-restoring forces for more complex moving parts such as regular convex and non-convex polygons and regular polygons with regular polygonal cavities. A closed-form approximate expression is derived for each of these polygonal geometries; this expression agrees with the numerical results obtained with the Surface Evolver software. For small shifts, it is found that the restoring force does not depend on the shift direction or on the polygonal shape. In order to tackle the problem of microsystem packaging, an extension of the theory is done for polygonal shapes pierced with connection vias (channels) and a closed form of the shift-restoring force is derived for these geometries and again checked against the numerical model. In this case, the restoring force depends on the shift direction. Finally, a non-dimensional number, the shift number, is proposed that indicates the isotropic or anisotropic behavior of the chip according to the shift direction.

F. Wakai, K. A. Brakke, Tensor virial equation of evolving surfaces in sintering of aggregates of particles by diffusion Acta Materiala 61 (2013) 4103-4112.
PDF file, 2.7 MB

Abstract: The moment of inertia tensor is a quantity that characterizes the morphology of aggregates of particles. The deviatoric components indicate the anisotropy of the aggregate, and its compactness is described by the isotropic component, i.e. the second moment of inertia, which is related to the radius of gyration. The equation of motion of the moment of inertia tensor is proposed for the sintering and coalescence of crystalline particles by bulk diffusion and surface diffusion. Simulations of the evolution of aggregates of particles (linear chains, rings and branched chains) show that the aggregates become more compact and more isotropic structures, driven by the surface energy tensor or the surface force density. The tensor virial equation for diffusion is applicable also to evolution of pores, precipitates and inclusions embedded in a surrounding matrix.

J. Berthier, K. A. Brakke, S. Mermoz, C. Fretigny, L. Di Cioccio, L. Stabilization of the tilt motion during capillary self-alignment of rectangular chips, J. Sensors and Actuators A: Physical, 234, 180-187 (2015).
PDF file, 2.7 MB

Abstract: Capillary self-alignment (CSA) has emerged as a convenient technique to assemble solid objects. In thistechnique a liquid droplet forces a mobile solid plate or chip to align with its counterpart on a solid substrate. It has been widely investigated for applications such as 3D microelectronics and assembly of optical components. It is now thought that it could be a solution for surface mounting and packaging technologies. For 3D microelectronics, where square or rectangular chips are used, it has been found that amongst the four displacement modes, i.e. shift, twist, lift and tilt, only the tilt mode was unstable (not restoring). In particular, tilting of a floating square or rectangular chip may trigger a direct contactbetween the plate and the pad that impedes alignment. In this text, an analysis of the tilt mode is firstpresented. Second, it is demonstrated that tilt can be stabilized by incorporating specific geometrical features such as lyophilic bands patterned on the substrate.