Bubbles Behaving Like Solid Shells

From left to right: Yousra Timounay, Olivier Pitois, Florence Rouyer

Authors: Yousra Timounay¹, Olivier Pitois², Florence Rouyer²

Affiliation:
¹Physics Department, Syracuse University, United States,
²Université Paris-Est, Laboratoire Navier, UMR 8205 CNRS, ENPC ParisTech, IFSTTAR, France.

Isolating small volumes of gas or liquid in the form of bubbles or droplets is important for a wide range of applications (pharmaceutics, cosmetics…), and they are major elements in the production of emulsions, foams, sprays, etc. Advanced applications require such fluid objects having specific features such as high mobility on solid substrates and significant robustness. The mechanical properties of individual bubbles/drops strongly impact the behavior of the material they are part of.

Controlling the properties of bubbles/drops and developing methods to tune them is therefore important. Surfactants and macromolecules (such as proteins) -- adsorbed at fluid interfaces -- lower intern overpressure (also called Laplace pressure ΔP_cap) of bubbles and drops by lowering surface tension. Solid particles can be advantageously used to confer unequaled properties to the resulting "armored" bubbles and drops. Droplets coated by solid particles, known as liquid marbles, exhibit unusual features; they are highly mobile [1] and their drying rate can be significantly decreased [2].

The dissolution of "armored" bubbles can be stopped as well [3]. Moreover, droplets and bubbles can be stable in a non-spherical shape when covered by particles [4]. Solid particles adsorbed at the interface can also mechanically strengthen bubbles and drops, so the later can sustain slightly negative pressures, i.e. ΔP ∼ ΔP_cap, before collapsing [3,5,6,7]. Note that in these systems, particles are bounded to by a unique fluid-fluid interface.

In our recent article [8], we have shown that the same concept can be applied to air bubbles within an air environment. These particulate bubbles, that we call gas marbles, are delimited by a monolayer of solid particles confined in a thin liquid film, i.e. bounded by two interfaces. Collapse pressures we measured are ∼10 times greater than the one reported for liquid marbles, i.e. ΔP ∼ 10 ΔP_cap. Moreover, gas marbles can sustain a significant overpressure before inflating. We attribute these remarkable results to the cohesion between the grains that form the shell delimiting gas marbles. Indeed, observation of the shell revealed that the liquid distributes in the pores between the particles (see Fig 1). This capillary structure is likely to induce attractive interactions between the particles. Thus, the remarkable strength of gas marbles results from the cohesion in the particulate shell.

Fig. 1 (a) is a side picture of a gas marble with 590 µm particles. (b) is a close-up showing liquid menisci between the particles. (c) and (d) are schemas showing the structure of a gas marble. Adapted from [8]

Gas marbles can be generated by immersing and withdrawing a rectangular frame into a surfactant solution with a particle raft laying on its interface. The particulate interface initially -- lifted by the frame -- eventually detaches due to its weight and closes over itself creating a gas marble. This technique is similar to how soap bubbles are created; in our case gravity replaces blowing onto the film to deform it. The liquid used to create gas marbles is an aqueous solution of Sodium Dodecyl Sulfate (SDS) with a surface tension equal to 36 mN/m. The particles are monosized polystyrene beads. Three different particle diameters were used: 250, 315 and 590 µm. To modify their wetting properties, the particles have been immersed into a mixture made up of octane solution and dissolved silane. After the silanization, the advancing contact angle between the particles and the SDS solution is equal to 95 ± 10^o making them slightly hydrophobic. Pictures and schemes showing the structure of gas marbles are presented in Fig 1. The diameter of gas marbles we study ranges between 5 to 12 mm.

We have characterized the properties of gas marbles by studying their behavior when submitted to pressure variations. In practice, we connect a gas marble to a syringe and a pressure sensor, by changing the volume of the syringe using a syringe pump, we induce either inflation or deflation of the gas marble under study. Increasing the syringe volume results in a decrease of the gas marble inner pressure, and vice versa. Thanks to two cameras placed above and below, we were able to follow the shape of gas marbles during experiments.

We have observed the same qualitative behavior during all experiments. First, the inner relative pressure of gas marbles is equal to zero at equilibrium (no pressure variation imposed). This result shows that -- contrary to soap bubbles -- Laplace’s law does not apply to gas marbles. Such effect results from the surface stress supported by the force network arising from contacts between particles trapped in the monolayer discussed above [9,10,11]. Second, during both inflation and deflation experiments, gas marbles go through a first regime where they do not deform despite the imposed overpressure/under-pressure. Eventually gas marbles deform at a critical relative pressure. A fracture consisting in the stretching of the liquid connecting the particles is observed at the critical overpressure while gas marbles collapse at the critical under-pressure. Critical relative pressures we measured do not seem to depend on the diameter of gas marbles and the size of the particles composing the shell within the size range we explored. Moreover, critical relative pressures have the same amplitude (∼ 500 Pa) for inflation and deflation experiments.

As said before, we attribute these exceptional properties to the particle confinement imposed by the double interface delimiting gas marbles, whereas a single interface is present for armored bubbles and drops. Concerning applications, gas marbles could be used to improve the stability of particle laden foams and emulsions. They could also be used to store and transport small volumes of gas.

References:
[1] Pascale Aussillous, David Quéré, "Liquid marbles". Nature, 411, 924 (2001). Abstract.
[2] Benoît Laborie, Florent Lachaussée, Elise Lorenceaua, Florence Rouyer, "How coatings with hydrophobic particles may change the drying of water droplets: incompressible surface versus porous media effects". Soft Matter, 9, 4822 (2013). Abstract.
[3] Manouk Abkarian, Anand Bala Subramaniam, Shin-Hyun Kim, Ryan J. Larsen, Seung-Man Yang, Howard A. Stone, "Dissolution arrest and stability of particle-covered bubbles". Physical Review Letters, 99, 188301 (2007). Abstract.
[4] Anand Bala Subramaniam, Manouk Abkarian, L. Mahadevan, Howard A. Stone, "Colloid science : Non-spherical bubbles". Nature, 438, 930 (2005). Abstract.
[5] O. Pitois, M. Buisson, X. Chateau, "On the collapse pressure of armored bubbles and drops". The European Physical Journal E, 38, 48 (2015). Abstract.
[6] Nicolas Taccoen, François Lequeux, Deniz Z. Gunes, Charles N. Baroud, "Probing the mechanical strength of an armored bubble and its implication to particle-stabilized foams". Physical Review X, 6, 011010 (2016). Abstract.
[7] Cécile Monteux, John Kirkwood, Hui Xu, Eric Jung, Gerald G. Fuller, "Determining the mechanical response of particle-laden fluid interfaces using surface pressure isotherms and bulk pressure measurements of droplets". Physical Chemistry Chemical Physics, 9(48), 6344-6350 (2007). Abstract.
[8] Yousra Timounay, Olivier Pitois, Florence Rouyer, "Gas Marbles: Much Stronger than Liquid Marbles". Physical Review Letters, 118, 228001 (2017). Abstract.
[9] Antonio Stocco, Wiebke Drenckhan, Emanuelle Rio, Dominique Langevina, Bernard P. Binks, "Particle-stabilised foams: an interfacial study". Soft Matter, 5, 2215 (2009). Abstract.
[10] Valeria Garbin, John C. Crocker, Kathleen J. Stebe, "Nanoparticles at fluid interfaces: Exploiting capping ligands to control adsorption, stability and dynamics". Journal of colloid and interface science, 387, 1 (2012). Abstract.
[11] Guillaume Lagubeau, Antonella Rescaglio, Francisco Melo, "Armoring a droplet: Soft jamming of a dense granular interface". Physical Review E, 90, 030201 (2014). Abstract.

On The Quest of Superconductivity at Room Temperature

Authors: Christian E. Precker¹, Pablo D Esquinazi¹, Ana Champi², José Barzola-Quiquia¹, Mahsa Zoraghi¹, Santiago Muiños-Landin¹, Annette Setzer¹, Winfried Böhlmann¹, Daniel Spemann^3,6, Jan Meijer³, Tom Muenster⁴, Oliver Baehre⁴, Gert Kloess⁴, Henning Beth⁵

Affiliation:
¹Division of Superconductivity and Magnetism, Institut für Experimentelle Physik II, Universität Leipzig, Germany,
²Centro de Ciências Naturais e Humanas, Universidade Federal do ABC, Santo André, São Paulo, Brazil,
³Division of Nuclear Solid State Physics, Institut für Experimentelle Physik II, Universität Leipzig, Germany,
⁴Institut für Mineralogie, Kristallographie und Materialwissenschaft, Fakultät für Chemie und Mineralogie, Universität Leipzig, Germany,
⁵Golden Bowerbird Pty Ltd., Mullumbimby, NSW, Australia,
⁶Present address: Leibniz Institute of Surface Modification, Physical Department, Leipzig, Germany.

Superconductivity is the phenomenon in nature where the electrical resistance of a conducting sample vanishes completely below a certain temperature which is known as “critical temperature (T_c)”. For low enough applied magnetic fields upon sample geometry, the phenomenon of flux expulsion (the Meissner effect) is observable, an effect of special importance for the physics of superconductivity. Due to its interesting characteristics, the phenomenon of superconductivity discovered by Kammerlingh Onnes in Leiden in 1911, is one of the most studied phenomena in experimental and theoretical solid state physics. It has important applications, like the generation of high magnetic fields using superconducting solenoids cooled at liquid He (4K) up to liquid nitrogen (77K) temperatures, or the use of extremely sensitive magnetic field sensors via the so-called Josephson effect. The higher the critical temperature, the easier is the use of superconducting devices, especially in microelectronic. The critical temperature of superconducting materials ranges between a few tens of mK to ~200K, this last critical temperature found recently in SxHy under very high pressures [1].

Among experts in low-temperature physics, in particular those with solid backgrounds on superconductivity, there exists a kind of unproven law regarding the (im)possibility to have superconductivity at room temperature, which means having a material with a critical temperature above 300K. In short, for most of the experts it is extremely difficult to accept that a room temperature superconductor would be possible at all, although there is actually no clear theoretical upper limit for T_c. This general (over)skepticism is probably the reason why, for more than 35 years, the work of Kazimierz Antonowicz [2] (on the superconducting-like behavior he observed on annealed graphite/amorphous carbon powders at room temperature [3]) was not taken seriously by the scientific community. Probably, the lack of easy reproducibility of the observed superconducting-like behavior and the vanishing of the amplitude of the signals within a few days [3] (added to the (over)skepticism of scientists) did not encourage them to look more carefully at those results. The work of Antonowicz on the room temperature superconductivity in carbon powders [3] was not cited in reviews discussing the possibility to reach superconductivity at room temperature, see, e.g., [4].

In the last 16 years, however, different measurements done in highly oriented pyrolytic graphite samples and graphite powders, see [5,6] for reviews, suggest that some kind of interfaces in the graphite structure may quite possibly be the origin for some of the measured signals. This may explain several aspects of this hidden superconductivity, like low reproducibility, time instability, small amount of superconducting mass and the difficulty to localize the superconducting phase(s).

Assuming that somewhere in graphite samples the room temperature superconductivity exists, the question arises: which is actually the critical temperature? This was the main question the work of Precker et al. [7] wanted to answer. For that purpose, the authors took natural crystals from Brazil and Sri Lanka mines. A reader would perhaps be surprised that in these days someone selects natural graphite crystals instead of highly pure and ordered pyrolytic graphite, so called HOPG, for research. The main reason to start with ordered natural crystals is that their several microns long interfaces are very well defined, see Fig. 1. The team in [7] also performed measurements with HOPG samples, whose results support those found in natural graphite crystals. Highly ordered natural graphite crystals of good quality were created during the earth's early evolution at temperature and pressure conditions unreachable in laboratories nowadays. Therefore, the well-defined stacking order phases (hexagonal and rhombohedral) and their interfaces shown in Fig.1 may contribute substantially to the metallic-like behavior of graphite [5,6].

Fig.1: (Click on the image to view with higher resolution) Scanning Transmission electron microscopy (STEM) pictures taken from three ~100nm thick lamellae from three different regions of a natural crystal from Brazil. The e-beam points always parallel to the graphene layers. The different colors mean different stacking ordered regions or regions with the same stacking order but rotated a certain angle around the c-axis. The c-axis is always normal to the graphene planes and interfaces. The picture (c) shows that there are regions in the same sample with no or much less interfaces density. The scale bars at the right bottom denotes 1 µm.

Detailed X-ray diffraction studies done in Ref.[7] show that in all samples a mixture of hexagonal (ABAB…, the majority phase) and rhombohedral (ABCABCA…) stacking orders exist in bulk graphite samples, independently of the sample origin. These two phases as well as their twist around the c-axis are the reason for the different colors in the STEM pictures of Fig.1. There are experimental [5,6] as well as theoretical reasons [8] that indicate that the origin for the metallic, and also most probably the superconducting behavior of graphite, is localized at some of those interfaces. One of the reasons why one expects superconductivity at certain interfaces, e.g. between rhombohedral and hexagonal stacking order, is that the relation between energy and wave-vector for conduction electrons becomes dispersionless. In this case and following the common BCS theory of superconductivity, the superconducting critical temperature is proportional to the Cooper pairs interaction strength. Therefore, it is expected that T_c is much higher than in the case of a quadratic dispersion relation [8].

Coming back to the main question, i.e. the critical temperature of the hidden superconductivity in graphite samples with interfaces, two results obtained in Ref.[7] and shown in Fig.2 resume the main evidence suggesting the existence of granular superconductivity below 350K in the measured crystal.

Figure 2(a) shows the temperature dependence of the resistance (a linear in temperature background is subtracted from the original data) around the transition. It is accompanied by the difference between the field cooled and zero field cooled magnetic moment that starts to increase at the lowest temperature onset of the transition in the resistance. Figure 2(b) shows the change in resistance for the same sample at 325K and after cooling it from 390K at zero field. The relatively large response of the resistance with field and the irreversibility are compatible with granular superconductivity; see also other results in [5,6].

Fig.2: (Click on the image to view with higher resolution) (a) The difference (left y-axis, red points) between the measured field cooled magnetic moment m_FC and the zero field cooled m_ZFC vs. temperature at a field of 50 mT applied at 250K for a natural graphite crystal from Brazil. Right y-axis: Difference between the measured resistance and a linear in temperature background vs. temperature -- for a sample from the same batch at zero field. (b) Change of the resistance with field at a temperature of 325K after cooling it from 390K at zero field. The field was applied normal to the interfaces.

The observed remanence in the resistance indicates that magnetic flux remains trapped within certain regions of the graphite samples. The origin and characteristics of this trapped flux and its non-monotonous temperature behavior [7] have to be clarified in the future and using other experimental techniques. One should also clarify to what extent a magnetically ordered state could have some influence on the observed phenomena. The observed phenomenology in Ref.[7] (see Fig.2) as well as in different studies done on graphite in the past [5,6] strongly suggest the existence of superconductivity. Although several details of the phenomenology, especially the large magnetic anisotropy of the effects in resistance, do not support magnetic order as a possible origin, one should not rule out yet the existence of unusual magnetic states at the graphite embedded interfaces, which are partially being studied theoretically nowadays, see, e.g., Ref.[9].

References:
[1] A. P. Drozdov, M. I. Eremets, I. A. Troyan, V. Ksenofontov, S. I. Shylin, "Conventional superconductivity at 203 kelvin at high pressures in the sulfur hydride system", Nature, 525, 73–6. Abstract.
[2] Kazimierz Antonowicz (1914–2003) started in the 60s the carbon research at Nicolas Copernicus University (Torum, Poland) investigating the structural and electronic properties of different forms of carbon.
[3] K. Antonowicz, "Possible superconductivity at room temperature", Nature, 247, 358–60 (1974). Abstract;   "The effect of microwaves on DC current in an Al–carbon–Al sandwich", Physica Status Solidi (a), 28, 497–502 (1975). Abstract.
[4] Arthur W. Sleight, "Room Temperature Superconductors", Accounts of Chemical Research, 28, 103-108 (1995). Abstract.
[5] Pablo Esquinazi, "Invited review: Graphite and its hidden superconductivity", Papers in Physics, 5, 050007 (2013). Abstract.
[6] P. Esquinazi, Y.V. Lysogorsky, "Experimental evidence for the existence of interfaces in graphite and their relation to the observed metallic and superconducting behavior", ed. P Esquinazi (Switzerland: Springer) pp 145-179 (2016), and refs. therein.
[7] Christian E Precker, Pablo D Esquinazi, Ana Champi, José Barzola-Quiquia, Mahsa Zoraghi, Santiago Muiños-Landin, Annette Setzer, Winfried Böhlmann, Daniel Spemann, Jan Meijer, Tom Muenster, Oliver Baehre, Gert Kloess, Henning Beth, "Identification of a possible superconducting transition above room temperature in natural graphite crystals", New Journal of Physics, 18, 113041 (2016). Abstract.
[8] T.T Heikkilä, G.E. Volovik, "Flat bands as a route to high-temperature superconductivity in graphite", ed. P Esquinazi (Switzerland: Springer) pp 123-143 (2016), and refs. therein.
[9] Betül Pamuk, Jacopo Baima, Francesco Mauri, Matteo Calandra, "Magnetic gap opening in rhombohedral stacked multilayer graphene from first principles", arXiv:1610.03445 [cond-mat.mtrl-sci].

Experimental Simulation of the Exchange of Majorana Zero Modes

From left to right: (top row) Jin-Shi Xu, Kai Sun, Yong-Jian Han; (bottom row) Chuan-Feng Li, Jiannis K. Pachos, and Guang-Can Guo.

Authors: Jin-Shi Xu^1,2, Kai Sun^1,2, Yong-Jian Han^1,2, Chuan-Feng Li^1,2, Jiannis K. Pachos³, Guang-Can Guo^1,2

Affiliation:
¹Key Laboratory of Quantum Information, Department of Optics and Optical Engineering, University of Science and Technology of China, China,
²Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, China,
³School of Physics and Astronomy, University of Leeds, UK.

The exchange character of identical particles plays an important role in physics. For bosons, such an exchange leaves their quantum state the same, while a single exchange between two fermions gives a minus sign multiplying their wave function. A single exchange between two Abelian anyons gives rise to a phase factor that can be different than 1 or -1, that corresponds to bosons or fermions, respectively. More exotic exchanging character are possible, namely non-Abelian anyons. These particles have their quantum state change more dramatically, when an exchange between them takes place, to a possibly different state. Such non-Abelian anyons are the Majorana fermions that were first proposed by the physicist Majorana [1].

Majorana zero modes (MZMs) are quasiparticle excitations of topological phases of matter that have the same exchange character of the Majorana fermions, that is, they are non-Abelian anyons. When two MZMs are exchanged, the process cannot be described only by a global phase multiplying their wave function. Instead, their internal states, corresponding to degenerate ground states of the topological system, are transformed by a unitary operator. In addition, states encoded in the ground-state space of systems with MZMs are naturally immune to local errors. The degeneracy of the ground state, driven by the corresponding Hamiltonian, cannot be lifted by any local perturbation. Therefore, the encoded quantum information is topologically protected. These unusual characteristics imply that MZMs may potentially provide novel and powerful methods for quantum information processing [2]. Rapid theoretical developments have greatly reduced the technological requirements and made it possible to experimentally observe MZMs. However, until now, only a few positive signatures of the formation of MZMs have been reported in solid-state systems. The demonstration of the essential characteristic of non-Abelian exchange and the property of topological protection of MZMs is a considerable challenge.

Recently, we use a photonic quantum simulator to experimentally investigate the exchange of MZMs supported in the 1D Kitaev Chain Model (KCM) [3]. The Fock space of the Majorana system is mapped to the space of the quantum simulator by employing two steps. First, we perform the mapping of the Majorana system to a spin-1/2 system via the Jordan-Wigner (JW) transformation. Then we perform the mapping of the spin system to the spatial modes of single photons. In this way, we are able to demonstrate the exchange of two MZMs in a three-site Kitaev chain encoded in the spatial modes of photons. We further demonstrate that quantum information encoded in the degenerate ground state is immune to local phase and flip noise errors.

We consider a three-fermion KCM which is the simplest model that supports isolated two MZMs. Six Majorana fermions are involved and the exchange of two isolated Majoranas can be realized by a set of projective measurement, which can be expressed as imaginary-time evolution (ITE) operators with a sufficiently large evolution time. These processes depend on the corresponding Hamiltonians. Figure 1 shows the exchanging process.

Figure 1: The exchange of Majorana zero modes. The spheres with numbers at their centers represent the Majorana fermions at the corresponding sites. A pair of Majorana fermions bounded by an enclosing ring represents a normal fermion. The wavy lines between different sites represent the interactions between them. The interactions illustrated in a, b, c and d represent different Hamiltonians, respectively. The figure is adapted from Reference [3].

We transformed the KCM to a spin model through the JW transformation. Although these two models have some different physics, they share the same spectra in the ferromagnetic region and their corresponding quantum evolution are equivalent. The geometric phases obtained from the exchanging evolution are invariant under the mapping. As a result, the well-controlled spin system offers a good platform to determine the exchanging matrix and investigate the exchange behavior of MZMs.

In our experiment, the states of three spin-1/2 sites correspond to an eight-dimension Hilbert space, which are encoded in the optical spatial modes of a single photon. To complete the exchange, we implement the ITE by designing appropriate dissipative evolution. The ground state information of the corresponding Hamiltonian is preserved but the information of the other states is dissipated. We use beam-displacers to prepare the initial states and the dissipative evolution is accomplished by passing the photons through a polarization beam splitter. In our case, the optical modes with horizontal polarization are preserved which represent the ground states of the Hamiltonian. The optical modes with vertical polarization are discarded.

Figure 2 shows the experimental results of simulating the exchanging evolution. States encoded in the two-dimension degeneracy space are represented in Bloch spheres. The final states (Figure 2b) after the exchanging evolution are obtained by rotating the initial states (Figure 2a) counterclockwise along the X axis through an angle of π/2. We obtain the exchanging matrix through the quantum process tomography [4]. The real and imaginary parts of the exchanged operator are presented in Figures 2c and d. Compare with the theoretical operation, the fidelity is calculated to be 94.13±0.04%.

Figure 2: Experimental results on simulating the exchanging evolution. a. The six experimental initial states in the Bloch sphere. b. The corresponding experimental final states after the braiding evolution. The final states are shown to be rotated along the X axis by π/2 from the initial states. c. Real (Re) and d. Imaginary (Im) parts of the exchange operator with a fidelity of 94.13±0.04%. The figure is adapted from Reference [3].

Figures 3a and b show the real and imaginary parts of the flip-error protection operator with a fidelity of 97.91±0.03%. Figures 4c and d show the real and imaginary parts of the phase-error protection operator with a fidelity of 96.99±0.04%. The high fidelity reveals the protection from the local flip error and phase error of the information encoded in the ground state space of the Majorana zero modes. The total operation behaves as an identity, thus demonstrating immunity against noise.

Figure 3: Experimental results on simulating local noises immunity. a. Real (Re) and b. Imaginary (Im) parts of the flip-error protection operator, with a fidelity of 97.91±0.03%. c. Real (Re) and d. Imaginary (Im) parts of the phase-error protection operator, with a fidelity of 96.99±0.04%. The high fidelity reveals the protection from the local flip error and phase error of the information encoded in the ground state space of the Majorana zero modes. The figure is adapted from Reference [3].

In our experiment, the optical quantum simulator provides a versatile medium that can efficiently simulate the Kitaev chain model that supports MZMs at its endpoints. It also opens the way for the efficient realization and manipulation of MZMs in complex architectures. The gained know-how can be picked up by other technologies that offer scalability, like ion traps or optical lattices. This work achieves the realization of non-Abelian exchanging and may provide a novel way to investigate topological quantities of quantum systems.

References:
[1] Ettore Majorana, "Symmetrical theory of electrons and positrons", Nuovo Cimento 14, 171 (1937). Abstract.
[2] Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, Sankar Das Sarma, "Non-Abelian anyons and topological quantum computation". Review of Modern Physics, 80, 1083 (2008). Abstract.
[3] Jin-Shi Xu, Kai Sun, Yong-Jian Han, Chuan-Feng Li, Jiannis K. Pachos, Guang-Can Guo, "Simulating the exchange of Majorana zero modes with a photonic system". Nature Communications", 7, 13194 (2016). Abstract.
[4] J. L. O'Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, A. G. White, "Quantum process tomography of a Controlled-NOT gate". Physical Review Letters, 93, 080502 (2004). Abstract.