FY2018 Annual Report

Mathematics, Mechanics, and Materials Unit
Professor Eliot Fried



FY2018 marked the fifth year since the Mathematics, Mechanics, and Materials Unit was established (originally under the moniker Mathematical Soft Matter Unit). An external review of the unit was conducted in mid-July of 2018. In its report to President Gruss, the review panel rated unit research contributions related to unstretchable material surfaces, the Kirchhoff–Plateau problem, and mechanical mechanisms and deployable structures as outstanding ("world-leading'") and characterized the unit leader as "a world-class applied mathematician in the field of mechanics". During FY2018, unit members published 6 peer-review articles, gave 11 colloquium or seminar presentations at other institutions and 7 invited presentations at international conferences and workshops, 3 contributed talks at international conferences, 3 poster presentations at international conferences, and applied for 2 patents. The unit leader accepted an invitation to join the editorial board of Mechanics of Materials (established 1982).

1. Staff

As of March 31, 2019

  • Dr. Eliot Fried, Professor
  • Dr. Johannes Schoenke, Staff Scientist
  • Dr. Nicolas Moreno Chaparro, Postdoctoral Scholar
  • Dr. Vikash Chaurasia, Postdoctoral Scholar
  • Dr. Alessandro Giussani, Postdoctoral Scholar
  • Dr. Stoffel Janssens, Postdoctoral Scholar
  • Dr. Adel Fernando Sarmiento Rodriguez, Postdoctoral Scholar
  • Dr. Burhannudin Sutisna, Postdoctoral Scholar
  • Dr. Alice Clair Tayler, Postdoctoral Scholar
  • Dr. David Vazquez Cortes, Research Unit Technician
  • Mr. Michael Grunwald, Research Unit Technician
  • Mr. Dmitrii Koldaev, Graduate Student
  • Mr. Alexandru Mihai, Graduate Student
  • Mr. Ali Rahmani, Graduate Student
  • Ms. Naoko Tokumoto, Research Unit Administrator


  • Dr. Tuan Hoang, Postdoctoral Scholar
  • Mr. Santiago Peña Clavijo, Special Research Student
  • Ms. Amanda Ho, Research Intern
  • Ms. Christina Nguyen, Research Intern
  • Ms. Clair Heffernan, Research Intern
  • Mr. Jake Chambrs, Research Intern

2. Collaborations

2.1 Unstretchable two-dimensional materials

  • Type of collaboration: Joint research
  • Researchers:
    • Professor Yi-Chao Chen, University of Houston, Houston, TX, USA
    • Professor Roger Fosdick, University of Minnesota, Minneapolis, MN, USA

2.2 Fabrication of diamond-based surface-acoustic-wave devices

  • Type of collaboration: Joint research
  • Researchers:
    • Professor Satoshi Fujii, Okinawa National College of Technology, Naha, Okinawa, JAPAN

3. Activities and Findings

3.1 Wrinkling instability of pressurized circular plates based on diamond thin films

Wrinkling is an ubiquitous phenomenon in nature and can be highly relevant to many applications, such as food packaging, energy harvesting, and mechanical testing. In this work, we investigate the wrinkling characteristics of circular diamond windows under compressive stresses and atmospheric pressure loads. The circular windows based on diamond thin films are prepared by the combination of chemical vapor deposition and lithography techniques. At an ambient pressure, the confined circular sheets show  localized wrinkles along the inner regime of its circumference. The extent of the wrinkled region is highly dependent on the window sizes. Although theoretical approaches have been demonstrated to explain similar wrinkling phenomena, e.g. as thin film blisters or floating sheets, our diamond windows exhibited unusual flat regimes at the center. More intriguingly, we find that the wrinkles characteristics deviate from the conventional buckling theory when inflating pressure loads are applied onto the  confined sheets. The video shows how the wrinkle pattern regularity increases along with the number of the wrinkles as the pressure increases. By combining theoretical analysis and experimental study, we propose a new theoretical model to describe the wrinkling instability of our diamond confined sheets under inflating pressures.


3.2  Interacting charged elastic loops on a sphere

The adsorption of polyelectrolytes on oppositely charged curved surfaces is of central importance in colloidal science and biophysics. Synthesis of  thin organic films are done by the stepwise deposition of polyelectrolyte chains on dissolvable spherical cores. Shells that remain after core dissolution have been used as containers for macromolecules, microcarriers, and microreactors. A more detailed theoretical framework is needed to gain further insight into the various factors governing the shape and stability of these chains. We use a variational approach to study the equilibrium configurations of a system consisting of two interacting curves that are constrained to lie on a sphere. Each loop is endowed with bending energy, a distance-dependent self-interaction energy, and a distance-dependent energy that accounts for interactions with the other loop. The first and second variation conditions are obtained in coordinate-free form. As an illustrative application, we consider a problem in which all interaction potentials are Coulombic, both the loops are of same length and bending rigidity and same charge density. We restrict this study to the case where the interaction is repulsive and the length of the loops is smaller than the length of the great circle of the sphere. Trivial equilibrium solution exist for such problem in the form of pair of parallel circular curves situated in opposite hemispheres. While the self-interaction stabilizes the trivial solution, the repul- sive inter-curve interaction has an opposite effect. We find that the dominant eigenmode which determines the stability of the trivial solution changes from first eigen mode to higher modes depending on the strength of the parameters of the system. This result is in contrast with problems involving only curvature energy where the lowest eigenmode is usually the dominant eigenmode as well.

Figure 3.2) Effect of inter-loop repulsion on 2-fold, 3-fold, and 4-fold equilibrium configurations.

3.3 Electrical conductance of thin lms on substrates with spherical
dents and bumps

To analytically estimate the electrical conductance of thin lms on substrates with spherical dents and bumps, the electrical conductance is calculated of thin lanes covering an insulating squared shaped substrate that contains a spherical cap with a base diameter equal to the length of the substrate. Exactly one insulating lane is present between two conductive lanes and all lanes appear to be straight lines from top view. For lanes that are relatively thin compared to the base, we nd that their parallel conductance is  17% less than that of lanes covering a at substrate. This nding allows us to estimate that the conductance of a thin lm of approximate uniform thickness on a substrate with spherical dents and bump is less than 17% below the conductance of a thin lm laying on a at substate.

Figure 3.3: a) Top view of a surface with a spherical cap that is endowed with conductive lane that are separated
by insulating lanes. One lane is visualized by a dotted line. The spherical cap has a base radius q, a height h,
and a radius of curvature r. b-c) Side views of a) including the sphere that forms the spherical cap.

3.4 3D structured diamond for neural tissue engineering applications

It is widely recognised that cells grown using a 3D cell culture technique more closely mimic natural tissues and organs than cells grown in 2D. In 2D culture, the adherence and spreading of cells on the flat surface is unnatural, compared to in 3D, where cells attach to one another and form natural cell-to-cell attachments. In this 3D cell culture environment, cells can exert forces on one another and can move and migrate as they do in-vivo. The benefits of 3D cell culture over 2D cell culture also manifest when mimicking pathological conditions such as cancer and neurodegenerative diseases including Alzheimer’s and Parkinson’s.

Here the initial goal is to establish a robust model in which neuronal cells spread evenly throughout a novel 3D diamond scaffold. Diamond has been shown to be an exceptional material for supporting the adhesion, proliferation and differentiation of neurons and Neural Stem Cells. A Lightfab 3D printer has been employed to create 3D structures in fused silica substrates which has then been used as a scaffold material for subsequent nanocrystalline diamond (NCD) film growths. Optimization of film quality was achieved via adapting growth conditions for the micro-structured fused silica scaffold.
The main benefits of using fused–silica–NCD as the base scaffold material is that the resulting material is optically transparent, facilitating the ease of imaging cells. The scaffold material is also robust and reusable, and initial cell culture results continue to highlight the advantage of exploiting diamond as a powerful tool for investigating the 3D neuronal networks.

Figure 3.4) Schematic of 3D diamond scaffold followed by optical images of a 5x5mm fused–
silica–diamond sample and SEM images (bottom row), highlighting the high resolution 3D
structuring and the high quality diamond film.

3.5 Interfacial fluid dynamics

Following the investigation of self propelled acetone droplets in a Leidenfrost state on liquid substrates by Janssens et al. a study of the dynamics with respect to the free surface of the liquid substrate was conducted. In this work we quanti ed the vertical displacement of the droplet relative to the equilibrium free surface. A model describing the motion of the droplet was also proposed. The results were presented at the Division of Fluid Dynamics APS meeting in the Gallery of Fluid Dynamics. From this work a proposed investigation of the deformation of a flowing free surface at low Reynolds numbers was formulated. Further experiments involving free surfaces in micro-channels, and equilibrium menisci around surface piercing cylinders of arbitrary cross sections. Along with experimental results a better theoretical understanding of the system is also being undertaken, as well as a computational approach using the Smooth Particle Hydrodynamics (SPH) model.

Figure 3.5)  Dynamics of an acetone drop in a Leidenfrost state on a 75oC water bath. Super-
imposed images taken at 40 frames per second.

3.6 Enhancement of texturing in diamond thin films grown by chemical vapor deposition

The space of deposition parameters of our microwave plasma chemical vapor deposition reactor has been investigated, aiming at optimizing the quality of the nanocrystalline diamond (NCD) films for the applications pursued by the group. One of the aspects of focus has been the enhancement of the degree of texturing of the diamond deposit, which tends to grow in random polycrystalline fashion. From this perspective the addition of a controlled amount of oxygen to the methane and hydrogen gas mixture proved to be beneficial at typically employed methane concentrations (3-5%). Fig. 3.6(a) depicts X-Ray Diffraction (XRD) scans of the diamond (111) Bragg peak for a series of samples of identical thickness (1.5 μm), deposited with a methane concentration of 5% for different oxygen concentrations. As [O₂] is raised, the intensity of the Bragg peak increases, pointing to a more strongly (111)-textured material. Figure 3.6(b) shows the Atomic Force Microscopy (AFM) images of the specimens. For zero/low [O₂],
grain enlargement is impeded by the formation of secondary nuclei which continuously give rise to new and differently oriented crystallites. This scenario results in weak and broad XRD reflections (black curve of Fig. 3.6(a)). At higher oxygen concentrations, oxygen and hydrogen radicals cooperate at the removal of non-diamond carbon from the surface of the growing deposit and suppress re-nucleation. Grains enlarge, grain boundaries containing undesired sp²-bonded carbon and hydrogen impurities become fewer, and the XRD peak in Fig. 3.6(a) gets sharper and more intense. It is noted that beyond a certain oxygen flow the growth rate slows down to impractical values and there is no further improvement of the diamond layer quality. The introduction of oxygen also leads to a reduction of the growth temperature. The results of this study will be implemented in the preparation of diamond thin films on glass for potential micromechanical/micro fluidic applications and on 3D fused silica structures for biological sensing, which require low growth temperature.

Figure 3.6) (a) Glancing incidence XRD 2θ scans around the diamond (111) Bragg peak as a function of [O₂] in
the gas mixture. (b) AFM images over (2x2)μm² for [O₂] = 0, 1, 2, and 3.5%. Color bars display topographic heights in nm.

3.7 Temporal stability of droplets distributed on an interface

The Cahn-Hilliard equation was proposed to model the phase separation of immiscible fluids [1]. More recently, it has been used to represent phenomena that involve interfacial dynamics such as wetting and tumor growth [2]. The multicomponent version of the Cahn-Hilliard model considers more than two phases and the corresponding interfaces between them. This study considers a system with three phases, specifically the stability of droplets that lie at the interface between two immiscible fluids.


3.8 Hierarchical assembly of triblock copolymer in solution

The assembly of soft-patchy nanoparticles emerges as a major breakthrough in the design functional and stimuli-responsive materials. The control of the morphology, mechanical and physicochemical stability of the assemblies is crucial to ensure the robustness of the fabrication process. In general, since the soft-patchy nanoparticles can deform (or undergo internal rearrangements), the complexity of their assemblies is higher than those obtained from the aggregation of rigid particles. Despite their potential in different industrial applications, the aggregation of soft-patchy nanoparticles is not well understood, and theoretical investigations are scarce. We use computational models investigate the effect of anisotropic interactions between nanoparticles and their capability to undergo morphological rearrangements. We model the aggregation of soft-patchy particles for various particles shape (see Figure 3.8.1) and thermodynamic conditions (interfacial and conformational entropy).

Fugure 3.8.1) Morphological variety of patchy particles used to investigate the formation of large aggregates

The morphology and growth rate of the mesoscopic aggregates can be then controlled to produce decorated structures with different characteristic size. In Figure 3.8.2 we compile different mesoaggregates morphologies obtained from various nanoparticles shapes. Additionally we present the alternative metastable morphologies that can be constructed as the entropic condition of the system change for a fixed nanoparticle shape. Figure 3.8.3 shows variety of aggregation mechanism that can be obtained for a fixed type of nanoparticles as the interfacial energy conditions change. The aggregation mechanism include stacking, merging, and combined staking-merging. Each mechanism is characterized by the topological and geometrical evolution of the aggregates.

Figure 3.8.2) Level of aggregation and morphologies obtained


Figure 3.8.3) Aggregation dynamic for three different interfacial entropic conditions. Three different aggregation mechanism were identified, leading to various characteristic size and morphologies of the final structure

3.9 Experimental and Computational Studies of the Sedimentation of Non-Orientable Objects

In many medical, industrial, and biological applications it is relevant to predict the sedimentation velocity of small particles under the gravity, and also how modifications in the shape of the objects can change the relative their relative motion. For example, traditional colloidal suspensions are comprised of symmetric objects (such as spheres, ellipsoids, cubes, etc.) in a fluid. However, modifying the morphology of suspended objects in a fluid offer unprecedented opportunities to tailor new properties in colloidal systems. The motion of objects with complex shape is richer and in general couples the translation and rotation of the object. Such richer responses can be exploited to fabricate steerable colloids, which orientation and transport can be controlled in much more detail.

We study experimental and computationally the sedimentation of rigid non-orientable shapes, also known as Möbius bands, constructed with one and three twists. We explore the effect of the geometrical features of this objects on their motion and the surrounding fluid. For the experimental prototype we investigate the sedimentation of 3D-printed polystyrene bands in water and water-glycerol mixtures. Figure 3.9.1 presents the experimental setup used to record the motion of the bands. Additionally, we use a particle-based discretization of the bands and the fluid to construct the computational models. For both, experimental and computational models we evaluate bands with different aspect ratios and orientation of the twist. We also compare the behavior of the Möbius bands with pi twist with their counterpart hollow cylinders (non-twisted band). We find that due to the geometrical features of these objects rotational and translational motions are coupled, giving rise to a complex sedimentation behavior describing helicoidal paths.

These objects exhibit a richer response in their motion which include translation, rotation, and tumbling. The tumbling frequency in both twist and non-twisted bands is controlled by the aspect ratio of object. The aspect ratio in the case of twisted bands determine the amplitude of the helicoidal trajectory (See video: SORtrajAllLine).

In the case of Möbius bands with three pi twist we compare two different schemes, using developable and nondevelopable surfaces. These objects resemble a propeler-like motion, exhibiting vertical translation as they sediment and rotation along their minor axis. For both schemes, the aspect ratio of the bands determines the rotation frequency.We identify a family of three twist bands which only the direction twist control the direction of the rotation (see Figure 3.9.3.a), whereas other constructions the aspect ratio of the band also influence the direction of the rotation, and can induce horizontal translation of the sedimenting object (Figure 3.9.3.b). For the family of bands in Figure 3.9.3.b we identify the critical aspect ratio where the rotation direction change.

Figure 3.9.1) Schematic representation of the Möbius band construction and experimental setup for single band sedimentation


Figure 3.9.2) Type of three twist Möbius bands investigated.


Figure 3.9.3) Dependency of the rotation direction on the twist direction and aspect ratio for the two different family of bands investigated.


4. Publications

4.1 Journals

  1. T. Hoang, E. Fried. Influence of a spanning liquid film on the stability and buckling of a circular loop with intrinsic curvature or intrinsic twist density, Mathematics and Mechanics of Solids 23 (2018), 43–66. (doi:10.1177/1081286516666135)
  2. G.G. Giusteri, E. Fried. Importance and effectiveness of representing the shapes of Cosserat rods and framed curves as paths in the special Euclidean algebra, Journal of Elasticity 132 (2018), 43–65. (doi:10.1007/s10659-017-9656-z)
  3. Y.-C. Chen, R. Fosdick, E. Fried. Issues concerning isometric deformations of planar regions to curved surfaces, Journal of Elasticity 132 (2018), 1–42. (doi:10.1007/s10659-017-9662-1)
  4. Y.-C. Chen, R. Fosdick, E. Fried. Isometric deformations of unstretchable material surfaces, a spatial variational treatment, Journal of the Mechanics and Physics of Solids 116 (2018), 290–322. (doi:10.1016/j.jmps.2018.03.020)
  5. S. Hartung, F. Sommer, S. Völkel, J. Schönke, I. Rehberg. Assembly of eight spherical magnets into a dotriacontapole configuration, Physical Review B, 98 (2018), 214424, (doi: 10.1103/PhysRevB.98.214424)
  6. J. Schönke, E. Fried. Single degree of freedom everting ring linkages with nonorientable topology, Proceedings of the National Academy of Sciences of the United States of America 116 (2019), 90–95. (doi: 10.1073/pnas.180976115)

4.2 Books and other one-time publications

Nothing to report

4.3 Oral and Poster Presentations

  1. E. Fried, Kaleidocycles and Möbius bands, Seminar, University of Hawai'i at Manoa, Hawaii, USA, April 27 (2018).
  2. T. Hoang, E. Fried, Stability of a magnetic ring subject to a point dipole, Conference Presentation, 10th European Solid Mechanics Conference (ESMC), Bologna, Italy, July 3 (2018).
  3. T. Hoang, E. Fried, Buckling of rings: A revisit, Seminar, University of Trento, Trento, Italy, July 9 (2018).
  4. E. Fried, Dimensionally reduced energy and equilibrium conditions for an isometrically deformed material surface, Seminar, Universitat Duisburg-Essen, Essen, Germany, July 23 (2011).
  5. J. Schönke, M. Grunwald, E. Fried, A Threefold Möbius Band with Constant Twist and Minimal Bending as the Limit of Tetrahedral Rings, Conference Presentation, Bridges 2018, Stockholm, Sweden, July 26 (2018).
  6. M. Grunwald, J. Schönke, E. Fried, Sevenfold and Ninefold Möbius Kaleidocycles, Conference Presentation, Bridges 2018, Stockholm, Sweden, July 26 (2018).
  7. J. Schönke, M. Grunwald, Theory, Design, and Construction of Mechanical Systems on a Tropical Island, Conference Presentation, The Bauhausinteraction Colloquium, Weimar, Germany, August 2 (2018).
  8. A. Taylor, P. Feretti, R. Jackman, E. Fried. Diamond as a material for stem cell regenerative medicine, Conference Presentation (invited), Elsevier International Conference on Diamond and Carbon Materials, Dubrovnik, Croaita, September (2018).
  9. A. Sarmiento, N. Moreno, E. Fried, Temporal stability of droplets distributed on an interface, Conference Poster Presentation, European Colloid and Iterface Society (ECIS) 2018, Liubljana, Slovenia, September 2 -- September 7 (2018).
  10. N. Moreno, Aggregation of soft-patchy colloids, Conference Presentation, European Colloid and Iterface Society (ECIS) 2018, Liubljana, Slovenia, September 11 (2018).
  11. J. Schönke, Oscillatory boundary dynamics coupled to a diffusive bulk, Conference Presentation, Theory of Biomathematics and Its Applications XV - Next Generation of Mathematical Sciences, Kyoto Japan, September 14 (2018).
  12. E. Fried, Building your own Möbius Kaleidocycles, Seminar, Illinois Geometry Lab, University of Illinois, Chicago, USA, September 17 (2018).
  13. E. Fried, Möbius kaleidocycles: a new class of everting ring linkages, Seminar, University of Illinois Urbana-Champaign, Champaign, USA, September 18 (2018).
  14. E. Fried, Möbius kaleidocycles, Seminar, American Institute of Mathematics, 600 E Brokaw Rd, SanJose, USA, October 9 (2018).
  15. E. Fried, Möbius kaleidocycles, Seminar, American Institute of Mathematics, Stanford University, Ventura Hall 220 Panama St, Stanford, USA, October 11 (2018).
  16. N. Moreno, D. Vasques-Cortes, E. Fried, Experimental and Computational Studies of the Sedimentation of Non-Orientable Objects, Conference Poster Presenatation, The SoR meeting, Houston, USA, October 14 -- October 18 (2018).
  17. E. Fried, Möbius kaleidocycles: a new class of everting ring linkages, Seminar, University of Southern California, Viterbi School of Engineering, The Laufer Library, USA, October 15 (2018).
  18. E. Fried, N. Moreno, Design of patchy nanoparticles via the self-assembly of triblock terpolymers in selective solvents, Conference Presentation, The 9th International Conference on Multiscale Materials Modeling, Osaka, Japan, October 31 (2018).
  19. S. Janssens, The effect of grain boundaries on the electronic transport properties of nanocrystalline diamond, Seminar,  Laboratoire des Sciences des Procédés et des Matériaux, CNRS, Université Paris 13, Sorbonne Paris Cité, Villetaneuse, France, December 8 (2018).
  20. E. Fried, Möbius kaleidocycles: a new class of everting ring linkages, Conference Presentation, 2018 Taiwan Mathematical Society Annual Meeting, National Taiwan Normal University, Taipei, Taiwan, December 8 (2018).
  21. E. Fried, Dynamics on membranes mediated by bulk diffusion, Conference Presentation, 2018 Taiwan Mathematical Society Annual Meeting, National Taiwan Normal University, Taipei, Taiwan, December 9 (2018).
  22. A. Taylor, A. Giussani, S. Janssens, 3D Structured diamond for neural tissue engineering applications, Conference Poster Presentation, Hasselt Diamond Workshop 2019-SBDD XXIV, Hasselt, Belgium, March 13 (2019).
  23. E. Fried, Single degree-of-freedom everting linkages with nonorientable topology, Seminar, University of Hawaii, Manoa, Hawaii, USA, March 25 (2019).
  24. E. Fried, The shape, energy, and stability of graphene Möbius bands, Conference Presentation, International Conference on Small Science (ICSS) 2019, Hawaii, USA, March 27 (2019).

4.4 Refereed Conference Publications

  1. J. Schönke, M. Grunwald, E. Fried, A Threefold Möbius Band with Constant Twist and Minimal Bending as the Limit of Tetrahedral Rings, Conference Presentation, Bridges 2018, Stockholm, Sweden, July 26 (2018). LINK
  2. M. Grunwald, J. Schönke, E. Fried, Sevenfold and Ninefold Möbius Kaleidocycles, Bridges 2018, Stockholm, Sweden, July 26 (2018). LINK


5. Intellectual Property Rights and Other Specific Achievements

Nothing to report

6. Meetings and Events

6.1 Seminars

Dr. Joe Welch (London Centre for Nanotechnology, University College London)

Dr. Andrés Baldelli (CNRS - Centre national de la recherche scientifique, France)

Professor Bob Svendsen (Professor of Material Mechanics, RWTH Aachen University, Germany)

Professor Jeffrey Giacomin (Professor of Chemical Engineering and of Mechanical and Materials Engineering at Queen’s University)

6.2 Events

6.2.1 Okinawa Future Semiconductor Technical Committee

  • Date: September 26, 2018
  • Venue: OIST Seminar room C209, Center building
  • Committee Chair: Prof. Fujii, N.I.T., Okinawa College
  • Sponsor: H30 Innovation Building Project in Okinawa Prefecture, and H30 KOSEN
                  Okinawa General Bureau, Cabinet Office, Prof. Fried OIST

7. Other

7.1 OIST Science Festival 2018

The unit provided interactive activities in which building Möbius Kaleidocycle.

7.2 OIST Science Challenge

Mr. Alexandru Mihai supported the science challenge as a study advisor, and provided a Lab tour.

7.3 Long Term Visiting Researcher

  • Professor Yi-chao Chen, University of Houston
  • Professor Fernando P. Duda, Federal University of Rio de Janeiro
  • Professor Gianni Royer-Carfagni, University of Parma
  • Professor Brian Seguin, Loyola University Chicago
  • Dr. James Kwiecinski