Minimum Surface Area Research Articles

Toward a Metallic Silicon Anode for Practical Lithium-Ion Battery Applications

The current state-of-the-art batteries comprising graphite-based anodes are facing its performance limit for the use in automotive applications, thus urging to develop electroactive materials with high capacity and low operating potentials [1]. Si has shown great promise as a next-generation anode, meeting the above requirements in natural abundance. However, unavoidable large volume change upon lithium insertion/extraction causes particle fracture and disintegration of the electrode, resulting in fast degradation of battery along with poor electric conductivity being the main hurdle for fast-chargeable battery design [2]. Recently, interesting breakthroughs have been proposed principally in nanostructuring of Si, composite formation with conductive carbons and mechanically stable binders, which achieved long-term cycle stability and rate capability [3]. Yet, these approaches lead to lowering the electrode density, initial Coulombic efficiency and energy density of battery and thus utilizing the Si microparticles can be rather feasible toward a practical system while stress-driven particle degradation becomes more serious. The existing technologies have focused on coalescing the fractured particles, not preventing itself, and have not considered ionic diffusion of Li-ion across the microparticles as well as the electronic conduction. Herein, we propose the material design concept of Si microparticles by integrating the chalcogen component into the bulk structure through a low temperature doping strategy. The low electric conductivity of intrinsic Si (~10-4 S m-1) with a near-insulator property undergoes a transition into the metallic state via this new method unlike other dopants of boron or phosphorous that only gives extra charge carriers without insulator-to-metal transition. In addition to electronic conduction, the chalcogen chains restrict the saturation of Si dangling bonding during recrystallization process of low temperature doping and create the internal channels inside the crystalline Si lattice. The incorporation of Li-ion channels further increases Li-ion diffusivity without any barriers in case of undoped Si microparticles due to the interfaces of lithium silicide and amorphous Si. Interestingly, chalcogen chains can sustain its internal structure with high flexibility and robustness which is directly corroborated by microscopy analysis, and lithium-induced intermediate can maintain the metallic nature at the interfaces that enhance the Li-ion diffusion over the cycles. Further, the porous but minimized surface area of Si structures increases the initial reversibility, extends the cycle life of battery up to hundreds of cycles with a high structural stability and facilitates fast-charging ability. [1] A. S. Arico, P. Bruce, B. Scrosati, J. M. Tarascon, W. Van Schalkwijk, Nat. Mater. 2005, 4, 366. [2] H. Wu, Y. Cui, Nano Today 2012, 7, 414. [3] J. Ryu, T. Chen, T. Bok, G. Song, J. Ma, C. Hwang, L. Luo, H.-K. Song, J. Cho, C. Wang, S. Zhang, S. Park, Nat. Commun. 2018, 9, 2924

An Upper Bound for the Smallest Area of a Minimal Surface in Manifolds of Dimension Four

In this paper, we prove that if M is a closed 4-dimensional Riemannian manifold with trivial first homology group, Ricci curvature $$|Ric|\le 3$$, diameter $${{\,\mathrm{diam}\,}}(M)\le D$$, and volume $${{\,\mathrm{vol}\,}}(M)>v>0$$, then the smallest area of a 2-dimensional minimal surface in M is bounded by F(v, D), for some function F that only depends on v and D. In order to prove this result, we first establish upper bounds for the first homological filling function of M that are of independent interest. This part of our work is based on recent results of Cheeger and Naber about manifolds with Ricci curvature bounded from both sides.

Ecological, fresh state and long-term mechanical properties of high-volume fly ash high-performance self-compacting concrete

In this work, the ecological efficiency, fresh state and long-term mechanical properties of high-volume fly ash (HVFA) high performance self-compacting concretes (SCCs) were evaluated. SCCs were prepared with 40–60% replacement of Portland cement by three fly ashes from two sources, with different finenesses. The rheological properties of the SCCs were evaluated by concrete workability tests. The efficiency of the fines to promote fresh state stability was also investigated. The compressive strength, modulus of elasticity and binder index were determined at 28, 91, 180 and 365 days. In addition, life-cycle assessment (LCA) analysis was performed to evaluate the influence of cement replacement by the fly ashes on the CO2 emission of concrete production. The results showed that cement replacement by fly ash reduced the viscosity and superplasticizer content of the SCCs, and improved their passing ability. The SCCs required a minimum surface area of fines of 120 × 106 m2/m3 of concrete to achieve fresh state stability. The finest fly ash obtained by grinding was the most efficient in promoting stability and resulted in the highest compressive strengths at all ages. The concretes showed significant compressive strength and modulus of elasticity increases up to 180 days. SCCs containing fly ash achieved a binder index of 5.8 kg.m−1.MPa−1 at 180 days and 5.6 at 365 days, lower than the 6.0 index of the reference at 28 days. The use of HVFA resulted in CO2-eq emissions (kg CO2-eq/m3 of concrete) and CO2-eq intensities (kg CO2-eq/MPa.m3 of concrete) up to 46 and 30% lower than the reference, respectively. Ground ash presented the best CO2-eq intensity indices, surpassing the additional energy spent to grind it.

Mixed micellization of a trisubstituted surface active ionic liquid 1-dodecyl-2,3-dimethylimidazolium chloride [C12bmim][Cl] with an amphiphilic drug amitriptyline hydrochloride AMT: A detailed insights from conductance and surface tension measurements

The thermodynamics and interfacial behavior of the mixed micellization of a trisubstituted imidazolium based surface active ionic liquid (SAIL), 1-dodecyl-2,3-dimethylimidazolium chloride [C12bmim][Cl], with an antidepressant amphiphilic drug amitriptyline hydrochloride (AMT) have been studied by employing surface tension measurements at 298.15 K and from conductance at 288.15, 298.15 and 308.15 K. Critical micelle concentration (cmc), degree of ionization (α), and other thermodynamic parameters like Gibbs free energy of micellization (∆Gmic.°), standard enthalpy of micellization (∆Hmic.°) and standard entropy of micellization (∆Smic.°) were derived from conductivity measurements. The respective surface active parameters like surface tension at cmc (γcmc), minimum surface area per ionic liquid molecule (Amin), adsorption efficiency (pC20), maximum surface excess concentration (Гmax), effectiveness of surface tension reduction (Πcmc), and cmc of the mixed systems of SAIL and AMT have been cognized using surface tension measurements. The negative values of ∆Gmic.° and the standard Gibbs free energy of adsorption at the air/water interface (∆Gad.°) indicate the feasibility of the micellization behavior of the mixed systems and confirm the feasibility of drug-SAIL interactions. The micellar interaction parameter (βm) have also been analyzed to get detailed insights of the interactions between the amphiphilic drug and the SAIL by applying theoretical model introduced by Rubingh and Motomura. The negative value of this parameter indicates the presence of synergistic interactions of SAIL-AMT binary mixtures. A comparative study has been done between disubstituted and trisubstituted SAILs and it has been found that the surface activity of trisubstituted SAIL is better than their disubstituted analogue due to the high hydrophobicity and hydrophilicity of trisubstituted SAIL.

Thermodynamic and spectroscopic studies on cationic surfactant tetradecyltrimethylammonium bromide in aqueous solution of trisubstituted ionic liquid 1, 2-dimethyl-3-octylimidazolium chloride at different temperatures

In this work, the effects on micellar behavior of long chain cationic surfactant tetradecyltrimethylammonium bromide (TTAB) upon the addition of trisubstituted ionic liquid (IL), 1, 2-dimethyl-3-octylimidazolium chloride [odmim][Cl] at temperatures, 298.15–318.15 K has been studied. Different techniques such as conductance, surface tension, fluorescence and 1H NMR have been employed to understand the interactional mechanisms. The values of critical micelle concentration (cmc) and various thermodynamic parameters have been calculated from conductivity measurements. The surface parameters like effectiveness of decrease in surface tension (Πcmc), minimum surface area occupied per surfactant monomer (Amin), maximum surface excess concentration (Γmax), and adsorption efficiency (pC20) have been evaluated by surface tension measurements. Micellar aggregation number (Nagg) has been determined by quenching of pyrene. Further to understand interactions in post micellar region, 1H NMR measurements have been performed. It has been observed that the lipophilicity of interacting ion modified the thermodynamic and aggregation properties of TTAB.

ISOMETRIC DEFORMATION OF THE SURFACE OF MINIMAL AREA OF A CYLINDER INCREASING ITS VOLUME

ISOMETRIC DEFORMATION OF THE SURFACE OF MINIMAL AREA OF A CYLINDER INCREASING ITS VOLUME Lizandro Baldomero, Reyna Zegarra y Heron Juan Morales Marchena Departamento de Matemáticas, Universidad Nacional del Santa, Chimbote, Perú DOI: https://doi.org/10.33017/RevECIPeru2010.0001/ ABSTRACT A method for increasing the volume of a cylinder with minimal area preserving the isometry is expounded. To do this, it was necessary to use numerical methods and a computer program. Such method let us set down a theorem which is analogy to that given for the case of a cube instead of a cylinder, being an immediate application of this theorem to the case of optimization in designing commercial bottles. To get an analytical proof of this new theorem is demanded. Keywords: surface of minimal area; optimization in designing commercial bottles; increasing the volume without stretching.

Aggregation Behavior of Polyether Based Siloxane Surfactants in Aqueous Solutions: Effect of Alkyl Groups and Steric Hindrance.

A series of polyether based siloxane surfactants with different branched chain and alkyl groups were synthesized by thiol-ene reaction and Piers-Rubinsztajn reaction. The effect of the siloxane structures (alkyl groups and branched chains) on the adsorption and aggregation behavior in aqueous solution was investigated by surface tension, fluorescence, dynamic light scattering (DLS), freeze-fracture transmission electron microscopy (TEM), and TEM. The molecular structures of siloxane can obviously influence their surface activities and thermodynamics. Replacing the methyl of trimethylsiloxyl groups with longer alkyl groups (ethyl, propyl, and butyl) and branching trimethylsiloxyl resulted in an obvious decrease of the values of critical micelle concentration (CMC) and surface tension at CMC (γCMC). Dense surface films packed with CH3 groups result in the lower surface tensions being disordered by longer alkyl groups or branched chains of siloxane hydrophobic groups. And the minimum surface area per surfactant molecule ( Amin) values of Si3-PG, Et-Si3-PG, Pro-Si3-PG, and But-Si3-PG successively decrease about 3.5 Å with each increasing -CH2- group. All polyether based siloxane surfactants can form nonuniform size spheroidal aggregates in aqueous solution. Concerning the driving force, the micellization process was spontaneous but less spontaneous compared with adsorption.

Estimating volume and surface area of a convex body via its projections or sections

The main goal of this paper is to present a series of inequalities connecting the surface area measure of a convex body and surface area measure of its projections and sections. We present a solution of a question from S. Campi, P. Gritzmann and P. Gronchi regarding the asymptotic behavior of the best constant in a recently proposed reverse Loomis-Whitney inequality. Next we give a new sufficient condition for the slicing problem to have an affirmative answer, in terms of the least outer volume ratio distance from the class of intersection bodies of projections of at least proportional dimension of convex bodies. Finally, we show that certain geometric quantities such as the volume ratio and minimal surface area (after a suitable normalization) are not necessarily close to each other.

Physicochemical Properties of Noncovalently Constructed Sugar-Based Pseudogemini Surfactants: Evaluation of Linker Length Influence

Natural renewable glucose or lactose upon reaction with dodecylamine has been converted into N-dodecylglucosylamine or N-dodecyllactosylamine, which upon acid–alkali reaction with dicarboxylic acid (HOOC(CH2)n-2COOH, n = 3,4,5,6,8) gives a series of sugar-based pseudogemini surfactants (G-n, L-n, respectively). Some physicochemical properties such as the critical micelle concentration (CMC), equilibrium surface tension at the CMC (γCMC), effectiveness (πCMC), efficiency (pC20), maximum surface excess (Γmax), minimum surface area (Amin), counterion binding of micelles (β), and the changes of standard Gibbs free energy (ΔGm0), enthalpy (ΔHm0) and entropy (ΔSm0) for processes of micellization in the range 298.15 K to 328.15 K have been evaluated by surface tension and electroconductometry methods in aqueous solutions of these pseudogemini surfactants. The results revealed that most of the above properties depend on dicarboxylic acid linker length and headgroup saccharide size. These findings help with unders...

Mixed micellization and surface properties of non-ionic/cationic surfactants

We investigate the surface properties of aqueous binary mixtures of our cationic surfactant O-dodecyl-N,N′-diisopropylisourea hydrochloride (ISO-DIC C12) with commercially available nonionic surfactant polyoxyethylene p-(1,1,3,3-tetramethylbutyl)phenyl ether (TritonX-100) at different temperatures (288 to 303 K). The micellization behavior of the binary systems is studied by determining the surface tension and other important physicochemical parameters, such as the critical micelle concentration (CMC), surface tension at the CMC(γcmc), Krafft Temperature (TK), maximum excess concentration (Γmax), minimum surface area per molecule (Amin), surface pressure at the CMC (Пcmc), and the adsorption efficiency (pC20) at the air/water interface. The study has additionally covered the calculation of thermodynamic parameters of micellization, including the standard Gibbs free energy, the standard enthalpy, the standard entropy, the free energy, and the Gibbs free energy of adsorption at air/water interface. The CMC values of the binary systems determined by experimental data are used to evaluate the micellar composition in the mixed micelle, the interaction parameter β and the activity coefficients f1(ISO-DIC C12) and f2 (polyoxyethylene p-(1,1,3,3-tetramethylbutyl)phenyl ether) using the theoretical treatment proposed by Clint and Rubingh. Our results reveal that the proposed binary systems possess enhanced surface activity compared to those of the individual surfactants.

Preprophase-band positioning in isolated tobacco BY-2 cells: evidence for a principal role of nucleus-cell cortex interaction in default division-plane selection.

In some plant tissue types, new cross-walls tend to divide parental cells equally and to meet parental walls at right angles while tending to have minimal surface area. A previously proposed model that I call the reach model suggests that this feature originates from the tendency of premitotic division-plane selection or of the positioning of microtubule preprophase bands (PPBs) which predict the cortical division site, and that default division-plane selection involves nuclear centering and subsequent PPB microtubule assembly on the cell wall parts closest to the nucleus. In an initial effort to characterize truly default division-plane selection, the present study quantified division orientation and PPB positioning in protoplast-derived isolated elongate tobacco BY-2 cells. In this system, PPB-predicted and actual division planes were mostly oriented transversely, as predicted based on the reach model. Some sample elongate cells had asymmetric shapes that came from clear terminal-size differences and, in those cells, PPB-marked planes tended to be displaced from the centers of centrally located nuclei toward the narrower cell end, again as predicted based on the reach model. Such PPB positioning typically forecasted volumetrically asymmetric transverse division that would produce a smaller daughter cell from a parental cell part including the narrower cell end. These results provide experimental evidence that default division-plane selection tends to be close to or the same as the selection using the reach model's criterion, and that it does not use any criterion that specifically prioritizes the equality or verticality of division.

Soluting effect of amino acids on 1‑decyl‑3‑methylimidazolium bromide and 1‑dodecyl‑3‑methylimidazolium bromide as cationic surfactants and sodium dodecyl sulfate as anionic surfactant in aqueous solutions

In this work, with the aim of shedding light on the effect of zwitterion cosolutes (amino acids) on the aggregation and surface behaviors of both cationic and anionic surfactants in aqueous solution, surface and micellar behavior of 1‑decyl‑3‑methyl imidazolium bromide ([C10mim]Br), 1‑dodecyl‑3‑methyl imidazolium bromide ([C12mim]Br) and sodium dodecyl sulfate (SDS) in aqueous solutions of 0.7 mol·kg−1 glycine, alanine and proline were studied by means of volumetric, compression, electric conductivity and surface tension measurements at different temperatures. From the experimental density and speed of sound data, the various parameters such as apparent molar volume, isentropic compression, apparent molar isentropic compression and the change of apparent molar volume upon micellization as well as upon transfer from pure water to aqueous amino acid solutions were derived. Our results showed that the order of isentropic compression for the investigated surfactants in presence of the amino acids was: SDS > [C12mim]Br ≫ [C10mim]Br. Also the obtained negative values for transfer apparent molar volume from pure water to aqueous amino acid solutions in ternary systems of ([C10mim]Br + water + proline) and ([C12mim]Br/SDS + water + glycine/alanine/proline) indicated that hydrophobic-hydrophobic interactions are predominant. The values of the degree of ionization of the counter ion on the micelles and thermodynamic properties of micellization for the investigated surfactants in aqueous amino acids solutions were obtained from the electrical conductivity measurements. It was found that, hydrophobic-hydrophobic interactions between the aliphatic chain of surfactants and amino acids have unfavorable effects on micelle formation. From the surface tension measurements, a series of parameters, including adsorption efficiency, surface tension at CMC, surface tension at micellar phase (γmic), effectiveness for surface tension reduction, maximum surface excess concentration and minimum surface area per molecule were determined for the investigated surfactants in the presence of different amino acids at 298.15 K.All the investigated amino acids have soluting-out effect and their abilities to reduce the CMC of the surfactants in aqueous solutions decrease with an increase in the hydrophobicity of the amino acid. Resultant data showed that the ability of the amino acids to decrease the CMC followed the order: glycine > alanine ≫ proline.

Hyperbranched Polyamidoamine Surfactants: Synthesis, Characterization and Evaluation as Biocides

AbstractDifferent generations of cationic hyperbranched quaternary ammonium polyamidoamine dendrimers (PAMAM) with methyl ester end group (G-0.5, G 0.5, G1.5 and G 2.5) were synthesized. Preparation of these dendrimers was carried out via alternative steps of an aza Michael addition reaction (of piperazine as core center molecule) and amidation reaction by ethylene diamine. Each aza Michael addition step was followed by a quaternization step using alkyl bromide with different chain length (4, 8 or 12) carbon atoms. The chemical structures of the prepared dendrimers were confirmed using FTIR, 1H-NMR spectra and elemental analysis for two generations as representative compounds. Also their surface activity has been studied and their surface parameters including surface and interfacial tension, emulsification power, critical micelle concentration, effectiveness, efficiency, maximum surface excess and minimum surface area were determined. The prepared quaternized dendrimers were tested as antimicrobial agents against different strains of bacteria, yeast and fungi. The results showed a significant antimicrobial efficacy of the synthesized surfactants.

Thermal design and optimization of a heat recovery steam generator in a combined-cycle power plant by applying a genetic algorithm

The heat recovery steam generator (HRSG) is an important and critical equipment of a combined cycle power plant that connects the gas turbine system to the steam cycle. The thermal design and the optimization of the HRSG are important for achieving safe operation, high efficiency and low product cost in a combined cycle power plant. This paper deals with the comprehensive optimization of the thermal design and minimization of the cost of an HRSG using a genetic algorithm (GA). Based on actual and existing designs of HRSG in most combined cycle power plants, a water tube HRSG including two superheaters, one evaporator and one economizer is considered in the optimization. A comprehensive program was developed in Visual Basic for this purpose.The optimization variables include the fin tube arrangement (transverse pitch, longitudinal pitch, number of rows in flue gas direction, number of tubes attached on circumference of headers, in line or staggered tube pattern), fin tube specification (tube diameter, fin height, fin thickness, fin type (solid or serrated), fin per meter, segment width of serrated fin) and also approach point, pinch point, water and steam velocity. On the other hand, the pressure at the exit of the gas turbine, fin tube metal temperature, amount of desuperheater spray water flow, steam pressure drops, minimum guaranteed thermal efficiency of HRSG, gap between fin tubes and the overall dimensions of HRSG (length, width, height) have been considered as main constraints in the optimization.The developed method selects and provides the best geometric parameter and arrangement of finned tubes based on minimum capital cost and relevant defined constraints. However, any other objective function such as minimum flue gas pressure drops, minimum heat transfer surface area, maximum rate of heat transfer, etc. could be used as objective functions in the program and easily optimized.In order to test this method, the results of the optimized design have been compared to an existing HRSG of a combined cycle power plant. All design parameters and the HRSG arrangement could be easily determined based on any optimization strategy and constraints.

A newly designed tympanostomy stent with TiO2 coating to reduce Pseudomonas aeruginosa biofilm formation

Biofilm formation has been implicated as a cause of post-tympanostomy tube otorrhea in patients suffering from otitis media with effusion, and biofilms have been found to adhere to all available types of tympanostomy tubes (TT) made from silicone. In this study, we present a novel stent designed with a reduced surface area and a titanium dioxide (TiO2) coating to prevent biofilm formation. Using a radio frequency power supply, tympanostomy stents (TS) made from Nitinol (Nikel-titanium) were coated with TiO2 to form an oxide layer on the metallic target. We successfully reproduced biofilms with carbenicillin-resistant Pseudomonas aeruginosa strain, PAO1-GFP (green fluorescent protein) on the tubes in vitro. We then compared the levels of biofilm formation by this strain on the two types of implants using several methods, including bacterial quantification, electron microscopy, and confocal laser fluorescent microscopy. Our results provide definitive evidence that the combination of the TiO2 coating and minimized surface area of the Nitinol stent inhibited the P. aeruginosa biofilm formation. The ability of the TS to prevent viable bacteria colonization (over 10 folds, compared to silicone TT) was verified by anti-biofilm test. Future studies will reveal more useful in reducing otorrhea and plugging complications as a novel tympanostomy tube.

Surface properties ofperfluorodecalin–containing liquid/liquid systems: The influence of Pluronic F-68 dissolved in the aqueous phase

Perfluorodecalin (PFD) is extensively applied as liquid carrier of gaseous metabolites in systems for in vitro cultures of various types of cells. The influence of Pluronic F-68 on modification of the surface properties of few liquid/liquid system containing PFD was discussed. Static surface and interfacial tensions (σ) of water/PFD, culture medium/PFD, as well as air/water (referentially) two-phases systems with the aqueous phase contained nonionic 3-block copolymer Pluronic F-68, were measured by the du Noüy ring method at 25 °C and 37 °C. Values of critical micelle concentration (CMC) for Pluronic F-68 dissolved in water/PFD and air/water systems were similar at constant temperature. Smaller value of maximum surface excess (Γ∞) and higher value of minimum surface area per adsorbed molecule (Amin) were obtained for water/PFD, than for water/air system. Increase of temperature from 25 °C to 37 °C resulted in significant decrease of CMC, decrease of Γ∞ and increase of Amin. Calculated value of the standard Gibbs free energy of micellization was negative and became more negative for higher temperature, what indicated that micelles were formed spontaneously and the tendency for micelle formation increased with temperature increase. The Frumkin model fits well the experimentally obtained values of σ, what has been proved by the coefficient of determination equaled to 0.994–0.996.

Synthesis of nanostructured titania/zirconia membrane and investigation of its physical separation and photocatalytic properties in treatment of textile industries wastewater

Nanostructured TiO2/ZrO2 composite membranes with varying compositions were obtained by sol–gel technique. The influence of 0–30 mol% zirconia doping on microstructure, water permeability, photocatalytic and physical separation properties, removal of methyl violet of textile industries wastewater and thermal and mechanical stability of titania/zirconia composite membranes was described. Firstly, alumina supports were coated with TiO2 intermediate layers using the colloidal sol–gel route. The TiO2/ZrO2 composite sols were prepared via a polymeric sol–gel method and dip-coated on TiO2 intermediate layer. The samples were characterized by DLS, TG-DTA, XRD, FTIR, BET-BJH, UV–visible, SEM, TEM and AFM. It was shown that zirconia retards the phase transformation of anatase to rutile until at least 700 °C. The minimum pore size and maximum surface area obtained were 1.2 nm and 153 m2/g, respectively, attributed to the sample with 20 mol% zirconia. The mechanical strength of titania membranes was significantly improved by addition of zirconia. The most methyl violet removal efficiency obtained, with and without UV-irradiation, is 80.8 and 72.6%, respectively, attributed to the sample with 20 mol% zirconia.

Sharp Area Bounds for Free Boundary Minimal Surfaces in Conformally Euclidean Balls

Abstract We prove that the area of a free boundary minimal surface $\Sigma ^2 \subset B^n$, where $B^n$ is a geodesic ball contained in a round hemisphere $\mathbb{S}^n_+$, is at least as big as that of a geodesic disk with the same radius as $B^n$; equality is attained only if $\Sigma $ coincides with such a disk. More generally, we prove analogous results for a class of conformally euclidean ambient spaces. This follows works of Brendle and Fraser–Schoen in the euclidean setting.

Inequalities for the Surface Area of Projections of Convex Bodies

AbstractWe provide general inequalities that compare the surface area S(K) of a convex body K in ℝn to the minimal, average, or maximal surface area of its hyperplane or lower dimensional projections. We discuss the same questions for all the quermassintegrals of K. We examine separately the dependence of the constants on the dimension in the case where K is in some of the classical positions or K is a projection body. Our results are in the spirit of the hyperplane problem, with sections replaced by projections and volume by surface area.

Holographic entanglement entropy in AdS4/BCFT3 and the Willmore functional

We study the holographic entanglement entropy of spatial regions having arbitrary shapes in the AdS4/BCFT3 correspondence with static gravitational backgrounds, focusing on the subleading term with respect to the area law term in its expansion as the UV cutoff vanishes. An analytic expression depending on the unit vector normal to the minimal area surface anchored to the entangling curve is obtained. When the bulk spacetime is a part of AdS4, this formula becomes the Willmore functional with a proper boundary term evaluated on the minimal surface viewed as a submanifold of a three dimensional flat Euclidean space with boundary. For some smooth domains, the analytic expressions of the finite term are reproduced, including the case of a disk disjoint from a boundary which is either flat or circular. When the spatial region contains corners adjacent to the boundary, the subleading term is a logarithmic divergence whose coefficient is determined by a corner function that is known analytically, and this result is also recovered. A numerical approach is employed to construct extremal surfaces anchored to entangling curves with arbitrary shapes. This analysis is used both to check some analytic results and to find numerically the finite term of the holographic entanglement entropy for some ellipses at finite distance from a flat boundary.