De Gennes Research Articles

Hölder regularity and convergence for a non-local model of nematic liquid crystals in the large-domain limit

We consider a non-local free energy functional, modelling a competition between entropy and pairwise interactions reminiscent of the second order virial expansion, with applications to nematic liquid crystals as a particular case. We build on previous work on understanding the behaviour of such models within the large-domain limit, where minimisers converge to minimisers of a quadratic elastic energy with manifold-valued constraint, analogous to harmonic maps. We extend this work to establish Hölder bounds for (almost-)minimisers on bounded domains, and demonstrate stronger convergence of (almost)-minimisers away from the singular set of the limit solution. The proof techniques bear analogy with recent work of singularly perturbed energy functionals, in particular in the context of the Ginzburg–Landau and Landau–de Gennes models.

Peng-Robinson Cubic Equation of State Based on Key Group Contribution and Calculation of Nitrogen Gas Solubility in MMA Dimer

Previously authors have reported facts that nitrogen (N2) and oxygen solubilities in styrene is the same as those in benzene and divinylbenzene at 303 K. Though the three compounds have an atomic composition, (CH)n (n=6, 8 and 10), the gas solubilities are thought to depend on the number of aromatic carbons in the solution. Then, authors named it ‘a key group in solubility’. In this research, the key group was investigated for N2 solubility in methyl methacrylate (MMA) and its dimer, ethylene glycol dimethacrylate (EGDMA). Therefore, the calculations were carried out by Peng-Robinson (PR) equation of state based on group contribution methods. The experimental data employed were those of Lai et al., where N2 solubility are reported in MMA at 303 K. The N2 solubility in EGDMA was assumed to be double of that in MMA at 303 K, because EGDMA has two MMA units in the molecule. Two types of group contribution methods were proposed to calculate N2 solubility in EGDMA. One was for the critical temperature and pressure proposed by Joback and Reid. Using these properties, the attractive and excluded volume parameters in PR equation were evaluated for EGDMA. Other was similar to a method proposed by Orbey and Sandler. The excluded volume parameter in PR equation can be evaluated from those of MMA by multiplying the number of repeating units, and the attractive by multiplying the squared number. Instead of the method, the powers were proposed to be 3/4 and 3/2 for the excluded volume and attractive perimeters, respectively. The powers were come from a scaling theory by de Gennes. The binary parameter for N2 – EGDMA was also set to k12 = 0.224, which was the same as that for N2 –MMA. The calculated N2 solubility in EGDMA was far smaller than the hypothetical N2 solubility in EGDMA at 303 K. Otherwise, to reproduce the hypothetical N2 solubility, a quite different value, k12 = -0.044, was necessary for the method by Joback and Reid. The results suggested that a key group for gas solubility will not be accepted in MMA.

Hydrodynamics of fringing-field induced defects in nematic liquid crystals

Consumer demand for high resolution and high refresh-rate displays has naturally led to the fabrication of liquid crystal displays with ever smaller pixels. As a consequence, fringing fields between adjacent pixels grow in magnitude, leading to abrupt changes in orientation. Electric field strengths above some threshold can lead to order melting and, in turn, disclinations. This paper presents accurate modeling of disclinations induced by fringing fields due to interdigitated electrodes in a nematic liquid crystal calculated by means of the Landau–de Gennes theory. Disclination paths are determined while taking into account the flow of the liquid crystal. Making use of interdigitated electrodes, precise electrical control over the creation and positioning of defects is demonstrated for homeotropic, planar, hybrid, and in-plane surface alignments.

Endothelial glycocalyx detection and characterization by means of atomic force spectroscopy: Comparison of various data analysis approaches

In recent years, atomic force spectroscopy (AFS) has been used to detect and characterize the endothelial glycocalyx (eGlx) in in vitro and ex vivo experiments. Several analysis methods were proposed, which differ not only in the numerical implementations, but also in physical models of glycocalyx description. Therefore, it is difficult to directly relate the experiments performed by different groups. In this work, we compared different models used for quantitative analysis of atomic force spectroscopy datasets recorded for eGlx. To capture glycocalyx at various structural conditions, we used basic enzymatic protocols for glycocalyx removal and restoration in human aortal endothelial cells (HAEC). Nanoindentation experiments for this model system were performed for (i) untreated cells, (ii) for cells after heparinase incubation, which enzymatically removes glycocalyx, (iii) for cells with successive heparin treatment, which partially restores the glycocalyx layer. Analysis of nanoindentation data was performed using different models: (a) a single-layer contact mechanics, (b) a double-layer model contact mechanics, (c) a polymer “brush” two-layer model based on the Alexander – de Gennes theory and (d) a simple single-layer “mechanical spring” model. Although different physical parameters are evaluated in methods (a–d), we show that all approaches revealed similar qualitative changes of the glycocalyx layer, which reflected the processes of glycocalyx degradation and its partial restoration. This paper may facilitate a direct comparison of past and future glycocalyx oriented AFS experiments that are analysed with different approaches.

Saturn ring defect around a spherical particle immersed in a nematic liquid crystal

We consider a nematic liquid crystal occupying the three-dimensional domain in the exterior of a spherical colloid particle. The nematic is subject to Dirichlet boundary conditions that enforce orthogonal attachment of nematic molecules to the surface of the particle. Our main interest is to understand the behavior of energy-critical configurations of the Landau–de Gennes Q-tensor model in the limit of vanishing correlation length. We demonstrate existence of configurations with a single Saturn-ring defect approaching the equator of the particle and no other line or point defects. We show this by analyzing asymptotics of energy minimizers under two symmetry constraints: rotational equivariance around the vertical axis and reflection across the horizontal plane. Energy blow-up at the ring defect is a significant obstacle to constructing well-behaved comparison maps needed to eliminate the possibility of point defects. The boundary estimates we develop to address this issue are new and should be applicable to a wider class of problems.

Solution landscapes of the simplified Ericksen–Leslie model and its comparisonwith the reduced Landau–deGennes model

We investigate the solution landscapes of a simplified Ericksen–Leslie (sEL) vector model for nematic liquid crystals, confined in a two-dimensional square domain with tangent boundary conditions. An efficient numerical algorithm is developed to construct the solution landscapes by utilizing the symmetry properties of the model and the domain. Since the sEL model and the reduced Landau–de Gennes (rLdG) models can be viewed as Ginzburg–Landau functionals, we systematically compute the solution landscapes of the sEL model, for different domain sizes, and compare them with the solution landscapes of the corresponding rLdG model. There are many similarities, including the stable diagonal and rotated states, bifurcation behaviours and sub-solution landscapes with low-index saddle solutions. Significant disparities also exist between the two models. The sEL vector model exhibits the stable solutionC±with interior defects, high-index ‘fake defect’ solutions, novel tessellating solutions and certain types of distinctive dynamical pathways. The solution landscape approach provides a comprehensive and efficient way for model comparison and is applicable to a wide range of mathematical models in physics.

Correlation of physical aging and glass transition temperatures in ultrathin polystyrene films supported on [formula omitted

This study examined the effects of the aging temperature (Tage) on structural relaxation in ultrathin-supported polystyrene (PS) films with thickness ~0.2Rg. X-ray reflectivity and differential scanning calorimetry were used to evaluate the glass transition temperature (Tg) of ultrathin PS and bulk PS, respectively. The ultrathin films in their confined state showed anomalous thickness relaxation behaviors that could be classified into three types: (1) Tage≤60°C, a slow increase in thickness with time and a decrease in relaxation magnitude with temperature; (2) 70°C≤Tage≤110°C, a slow decrease in thickness with time and an increase in relaxation magnitude with temperature; (3) Tage>110°C, an increase in thickness with time due to dewetting of PS films deposited on SiO2. Such an interesting relaxation behavior can arise from the enhanced dynamics at the surface and reduced mobility at the interface inherent in ultrathin-supported films. Two distinct Tgs peculiar to ultrathin films were confirmed. The lower Tg has not been reported for supported PS films. The relaxation summarized above revealed a strong correlation with the Tg values. The de Gennes modified sliding motion model was applied to explain the correlation with relaxation dynamics.

Identification of the Majorana edge modes in tight-binding systems based on the Krylov method

Low dimensional structures in the non-trivial topological phase can host the in-gap Majorana bound states, identified experimentally as zero-bias peaks in differential conductance. Theoretical methods for studying Majorana modes are mostly based on the bulk-boundary correspondence or exact diagonalization of finite systems via, e.g., Bogoliubov–de Gennes formalism. In this paper, we develop an efficient method for identifying the Majorana in-gap (edge) states via looking for extreme eigenvalues of symmetric matrices. The presented approach is based on the Krylov method and allows for study the spatial profile of the modes as well as the spectrum of the system. The advantage of this method is the calculation cost, which shows linear dependence on the number of lattice sites. The latter problem may be solved for very large clusters of arbitrary shape/geometry. In order to demonstrate the efficiency of our approach, we study two- and three-dimensional clusters described by the Kitaev and Rashba models for which we determine the number of Majorana modes and calculate their spatial structures. Additionally, we discuss the impact of the system size on the physical properties of the topological phase of the magnetic nanoisland deposited on the superconducting surface. In this case, we have shown that the eigenvalues of the in-gap states depend on the length of the system edge.

Nematic ordering of model racemic mixture of stiff trimer chains

The nematic ordering of (1) a racemic mixture and (2) an enantiopure melt of trimer chains is inspected within the Landau theory of phase transitions. Each chain contains three rigid segments; consecutive segments are connected at an external angle ν while the planes of the first two and the last two segments form the dihedral angle φ. It is assumed that the rigid segments comprise monomer units. The Landau–de Gennes expansion of the free energy of the melt has been obtained up to the sixth order in powers of the nematic order parameter, the coefficients of this expansion have been calculated from the microscopic model of trimer molecule. It is found that phase diagrams for a racemic mixture may contains the regions of stability of isotropic, prolate uniaxial, oblate uniaxial, and biaxial nematic phases depending on the values of angles ν,φ, and monomer unit fractions of rigid segments that characterize the architecture of a trimer chain. In turn, a chiral nematic states are found for an enantiopure melt, depending on the architecture of the chains.

Electrically Tuneable Optical Diffraction Gratings Based on a Polymer Scaffold Filled with a Nematic Liquid Crystal.

We present an experimental and theoretical investigation of the optical diffractive properties of electrically tuneable optical transmission gratings assembled as stacks of periodic slices from a conventional nematic liquid crystal (E7) and a standard photoresist polymer (SU-8). The external electric field causes a twist-type reorientation of the LC molecules toward a perpendicular direction with respect to initial orientation. The associated field-induced modification of the director field is determined numerically and analytically by minimization of the Landau–de Gennes free energy. The optical diffraction properties of the associated periodically modulated structure are calculated numerically on the basis of rigorous coupled-wave analysis (RCWA). A comparison of experimental and theoretical results suggests that polymer slices provoke planar surface anchoring of the LC molecules with the inhomogeneous surface anchoring energy varying in the range 5–20 μJ/m2. The investigated structures provide a versatile approach to fabricating LC-polymer-based electrically tuneable diffractive optical elements (DOEs).

Torus-like solutions for the Landau–De Gennes model.

In this note we report on some recent progress [10, 11, 12] about the study of global minimizers of a continuum Landau–De Gennes energy functional for nematic liquid crystals in three-dimensional domains. First, we discuss absence of singularities for minimizing configurations under norm constraint, as well as absence of the isotropic phase for the unconstrained minimizers, together with the related biaxial escape phenomenon. Then, under suitable assumptions on the topology of the domain and on the Dirichlet boundary condition, we show that smoothness of energy minimizing configurations yields the emergence of nontrivial topological structure in their biaxiality level sets. Then, we discuss the previous properties under both the norm constraint and an axial symmetry constraint, showing that in this case only partial regularity is available, away from a finite set located on the symmetry axis. In addition, we show that singularities may appear due to energy efficiency and we describe precisely the asymptotic profile around singular points. Finally, in an appropriate class of domains and boundary data we obtain qualitative properties of the biaxial surfaces, showing that smooth minimizers exibit torus structure, as predicted in [16, 24, 25, 39].

Selection rule for topological amplifiers in Bogoliubov–de Gennes systems

Dynamical instability is an inherent feature of bosonic systems described by the Bogoliubov de Geenes (BdG) Hamiltonian. Since it causes the BdG system to collapse, it is generally thought that it should be avoided. Recently, there has been much effort to harness this instability for the benefit of creating a topological amplifier with stable bulk bands but unstable edge modes which can be populated at an exponentially fast rate. We present a theorem for determining the stability of states with energies sufficiently away from zero, in terms of an unconventional commutator between the number conserving part and number nonconserving part of the BdG Hamiltonian. We apply the theorem to a generalization of a model from Galilo et al. [Phys. Rev. Lett, 115, 245302(2015)] for creating a topological amplifier in an interacting spin-1 atom system in a honeycomb lattice through a quench process. We use this model to illustrate how the vanishing of the unconventional commutator selects the symmetries for a system so that its bulk states are stable against (weak) pairing interactions. We find that as long as time reversal symmetry is preserved, our system can act like a topological amplifier, even in the presence of an onsite staggered potential which breaks the inversion symmetry.

Adsorption Kinetics of cis-1,4-Polyisoprene in Nanopores by In Situ Nanodielectric Spectroscopy

Using in situ nanodielectric spectroscopy, we studied the adsorption kinetics of cis-1,4-polyisoprene (PI) into porous alumina by following the evolution of the dielectrically active longest normal mode. We studied the influence of molar mass, nanopore diameter, and surface functionalization. Adsorption times depend strongly on the ratio 2Rg/D, where Rg is the radius is gyration and D is the pore diameter. For a given pore diameter, the characteristic adsorption times are some 8 orders of magnitude slower than the terminal relaxation times and more than 12 orders of magnitude slower than the segmental times. The extremely slow kinetics reflect the fact that exchanging chains with the pore surface have to pass through several unfavorable configurations (e.g., trains, loops). The molar mass dependence of the characteristic adsorption times (τads ∼ N2.6) is in good agreement with a scaling theory proposed by de Gennes and later refined by Semenov and Joanny. Subsequently, we investigated the imbibition of miscible PI blends by taking advantage of the difference in imbibition speeds of the respective homopolymers. We show that the shorter chains penetrate first the nanopores, whereas the longer chains enter only at the late stages of the filling process. Moreover, the long-time adsorption is dominated by an exchange mechanism involving primarily the shorter chains. The results from in situ nanodielectric spectroscopy demonstrate the capacity of the technique to provide the imbibition length, the adsorption kinetics, and, at the same time, the chain dynamics.

Microscopic derivation of superconductor-insulator boundary conditions for Ginzburg-Landau theory revisited: Enhanced superconductivity at boundaries with and without magnetic field

Using the standard Bardeen-Cooper-Schrieffer (BCS) theory, we revise microscopic derivation of the superconductor-insulator boundary conditions for the Ginzburg-Landau (GL) model. We obtain a negative contribution to free energy in the form of surface integral. Boundary conditions for the conventional superconductor have the form $\textbf{n} \cdot \nabla \psi = \text{const} \psi$. These are shown to follow from considering the order parameter reflected in the boundary. The boundary conditions are also derived for more general GL models with higher-order derivatives and pair-density-wave states. It shows that the boundary states with higher critical temperature and the boundary gap enhancement, found recently in BCS theory, are also present in microscopically-derived GL theory. In the case of an applied external field, we show that the third critical magnetic-field value $H_{c3}$ is higher than what follows from the de Gennes boundary conditions and is also significant in type-I regime.

Direct observation of the scaling relation between density of states and pairing gap in a dirty superconductor* *Project supported by the National Key Research and Development Program of the Ministry of Science and Technology of China (Grant Nos. 2017YFA0302903, 2016YFA0300602, 2016YFA0300604, and 2016YFJC010282), the National Natural Science Foundation of China (Grant Nos. 11227903, 12004416, and U2032204), the Beijing Municipal Science

Theories and experiments on dirty superconductors are complex but important in terms of both theoretical fundamentals and practical applications. These activities are even more challenging when magnetic fields are present because the field distribution, electron density of states, and superconducting pairing potentials become nonuniform. Here, we present tunneling microspectroscopic experiments on NbC single crystals and demonstrate that NbC is a homogeneous dirty superconductor. When applying magnetic fields to the samples, we found that the zero-energy local density of states and the pairing energy gap followed the explicit scaling relation proposed by de Gennes for homogeneous dirty superconductors in high magnetic fields. More significantly, our experimental findings indicate that the validity of the scaling relation extends to magnetic field strengths far below the upper critical field, calling for a new nonperturbative understanding of this fundamental property in dirty superconductors. On the practical side, we used the observed scaling relation to derive a simple and straightforward experimental scheme for estimating the superconducting coherence length of a dirty superconductor in magnetic fields.

Tricritical Phenomena and Cascades of Temperature Phase Transitions in a Ferromagnetic Liquid Crystal Suspension

We consider temperature-driven phase transitions occurring in a liquid crystal suspension of ferromagnetic particles within the Landau–de Gennes theory. The temperature dependences of the order parameters in the uniaxial model with a vector order parameter for the magnetic subsystem are obtained. The dimensionless expression for the free energy density of the suspension has been used for the study of the phase behavior general regularities of the system. Phase state diagrams of the suspension and temperature dependences of the order parameters of the liquid crystal and the ensemble of magnetic particles for different values of the phenomenological expansion coefficients are constructed. It is shown that the considered model admits the existence of a cascade of temperature phase transitions: isotropic phase–superparamagnetic nematic phase–ferromagnetic nematic phase. We have shown that in the mesomorphic state of the liquid crystal, the spontaneous magnetization can appear in a continuous way or by a jump with decreasing temperature, which corresponds to the tricritical behavior. The values of temperature and expansion coefficients corresponding to the tricritical and triple points are numerically found.

Liquid crystal droplets under extreme confinement probed by a multiscale simulation approach

ABSTRACT In this work, we computationally investigate liquid crystal (LC) droplets in the size range 0.03–1 μm, confined within shells of combined anchoring conditions. Two different types of surface were defined to promote homeotropic and planar degenerate anchoring, respectively. We identified the LC behaviour within the nanoscale droplets using a bespoke multiscale simulation approach. To study 30 nm droplets, we used coarse grained simulations within the dissipative particle dynamics formalism; to study 0.1 μm and larger droplets, we used a finite element method based on the Landau–de Gennes theory. Good agreement between the two methods was observed in our prior analysis and was confirmed in the present work. We explicitly study droplets of size 0.1 and 1 μm by using continuum mechanics calculations. Our results for the largest droplet are consistent with those available in the literature, suggesting that the extension to smaller droplets presented here is realistic, and therefore can be helpful for innovations in which device intensification could be achieved using LC nanodroplets.

Nanoscale dynamics during self-organized ion beam patterning of Si. II. Kr+ bombardment

Despite extensive study, fundamental understanding of self-organized patterning by broad-beam ion bombardment is still incomplete and controversial. Understanding the nanopatterning of elemental semiconductors, particularly silicon, is both foundational for the broader field and of intrinsic scientific and technological interest itself. This is the second component of a two-part investigation of the kinetics and fluctuation dynamics of self-organized nanoscale ripple development on silicon during 1 keV Ar$^+$ (Part I) and Kr$^+$ bombardment. Here, it's found that the ion-enhanced viscous flow relaxation is essentially equal for Kr$^+$-induced patterning as previously found for Ar$^+$ patterning. The magnitude of the surface curvature dependent roughening rate in the early stage kinetics is larger for Kr$^+$ than for Ar$^+$, qualitatively consistent with expectations for erosive and mass redistributive contributions to the initial surface instability. As with the Ar$^+$ case, fluctuation dynamics in the late stage show a peak in correlation time at the length scale corresponding to the dominant structural feature on the surface -- the ripples. Analogy is made to the phenomenon of de Gennes narrowing in liquids, but significant differences are also pointed out. Finally, it's shown that speckle motion during the surface evolution can be analyzed to determine spatial inhomogeneities in erosion rate and ripple velocity on the surface. This allows the direction and speed of ripple motion to be measured in real time, a unique capability for measuring these fundamental parameters outside the specialized environment of FIB/SEM systems.

Josephson current in an irradiated Weyl semimetal junction* *Project supported by the National Key Research and Development Program of China (Grant No. 2017YFA0303203) and the National Natural Science Foundation of China (Grant No. 11474149).

The influence of the off-resonant circularly polarized light on the Josephson current in the time-reversal broken superconducting Weyl semimetal junctions is investigated by using the Bogoliubov–de Gennes equation and the transfer matrix approach. Both the zero momentum BCS pairing states and the finite momentum Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) pairing states are considered for the Weyl superconductors. When a circularly polarized light is applied, it is shown that the current phase relation remains unchanged for the BCS pairing with the increasing of incident radiation intensity A 0. For FFLO pairing, the Josephson current exhibits the 0–π transition and periodic oscillation as a function of A 0. The dependence of free energy and critical current on A 0 are also investigated.

Open Problems in Wetting Phenomena: Pinning Retention Forces.

We review existing explanations for drop pinning and the origin of the force required to initiate the sliding of a drop on a solid surface (depinning). Theories that describe these phenomena include de Gennes', Marmur's, Furmidge's, the related Furmidge-Extrand's, and Tadmor's theory. These theories are all well cited but generally do not address each other, and usually papers that cite one of them ignore the others. Here, we discuss the advantages and disadvantages of these theories and their applicability to different experimental systems. Thus, we link different experimental systems to the theories that describe them best. We describe the force laws that can be deduced should these theories be united and the major open problems that remain. We describe a physical meaning that can be extracted from retention force measurements, specifically, the interfacial modulus that describes the tendency of a solid to conform to the liquid. This has implications for various wetting phenomena such as adhesion robustness, drug penetration into biological tissues, and solid robustness/resilience versus solid degradation over time as a result of its contact with a liquid.