## What is/are Isotropic Surface?

Isotropic Surface - The directional reflection characteristics of anisotropic surfaces play an important role in the precise modeling of radiative heat transfer and target detection.^{[1]}Gold nanoparticles (AuNPs) covered with mixtures of immiscible ligands present potentially anisotropic surfaces that can modulate their interactions at complex nano-bio interfaces.

^{[2]}We revisit the crack kinking problem in materials with strongly anisotropic surface energies by using a variational phase-field model.

^{[3]}Understanding the diffusion of adatom in anisotropic surface on the fc.

^{[4]}We present a novel anisotropic surface remeshing method that can efficiently eliminate obtuse angles.

^{[5]}First, based on the thermodynamic variation and smooth vector-field perturbation method, we rigorously derive a sharp-interface model with weakly anisotropic surface energies, and this model describes the interface evolution which occurs through surface diffusion flow and contact line migration.

^{[6]}A novel phase field material point method is introduced for robust simulation of dynamic fracture in elastic media considering the most general case of anisotropic surface energy.

^{[7]}An anisotropic surface pattern named polar microstructure is proposed to provide the global map for the matching.

^{[8]}For this purpose we study in situ and in real time, by Low Energy Electron Microscopy, the motion of 2D one-atom thick islands or one-atom deep holes on a highly anisotropic surface (reconstructed Si(1 0 0)).

^{[9]}Because of the hierarchical structure and anisotropic surface, an obvious red shift of the local surface plasmon resonance spectrum was observed for the branched Au nanoshells.

^{[10]}Results showed that all the Cr films had a strong (110) preferred orientation and anisotropic surface morphology with columnar structures.

^{[11]}The described limited crystal phase nucleation, the circumstances of which are unique in the realm of small-molecule organic semiconductors, is explained in terms of a two step nucleation mechanism while taking into account the anisotropic surface energy of crystal grains.

^{[12]}In this work, ice friction experiments with stainless steel samples that have different isotropic surface roughness values were carried out.

^{[13]}In this Letter, to the best of our knowledge, a novel anisotropic surface impedance holographic metasurface antenna for generating orbital angular momentum (OAM) is developed through design, fabrication, and measurement at radio frequencies.

^{[14]}The dynamics of interfacial evolution of a particle in the binary alloy melt affected by anisotropic surface tension is studied by means of the multiple variable expansion method.

^{[15]}An anisotropic surface modulation is observed above the substrate recrystallization temperature ∼300 °C.

^{[16]}Chiral molecular walkers, standing on their ‘feet’ on an anisotropic surface, perform preferential unidirectional Brownian motion under the influence of an external oscillating field according to their orientation, conformation and chirality.

^{[17]}To this end, a novel description of the local thermodynamic equilibrium of frozen water (ice) and a mixture of water vapour and an inert gas is derived, where anisotropic surface energy densities can be considered.

^{[18]}The broken bond density on the exposed surfaces of fluorite was calculated, enabling the anisotropic surface reactivity of fluorite such as surface energy, surface relaxation, dissolution, wettability, surface charge and adsorbability to be revisited.

^{[19]}Given an elliptic integrand of class $ \mathscr{C}^{3} $, we prove that finite unions of disjoint open Wulff shapes with equal radii are the only volume-constrained critical points of the anisotropic surface energy among all sets with finite perimeter and reduced boundary almost equal to its closure.

^{[20]}This paper studies the effect of anisotropic surface tension on the morphological stability of deep cellular crystal in directional solidification by using the matched asymptotic expansion method a.

^{[21]}Statistically significant differences in peak dimension were found between isotropic and anisotropic surfaces at scales of 15 μm and 20 μm.

^{[22]}When colloidal particles are placed into a liquid crystal host the anisotropic surface interactions produce spatial elastic distortions described as elastic multipoles.

^{[23]}For silicon, where experiments on grain growth exhibit {111} and {100} facets [103], we show how to obtain the interfacial energy to a given shape and thus obtain an appropriate anisotropic surface function.

^{[24]}We present a transmission-type transistor-based magnetless nonreciprocal metasurface that can properly be characterizable by bianisotropic surface susceptibilities.

^{[25]}For nickel, we observe the onset of anisotropic surface melting associated with each of the two surface facets present, but this gives way to uniform surface melting and the solid melts radially until the solid core eventually breaks up.

^{[26]}As applications, we provide descriptions of minimal surfaces in $${\mathbb {R}}^{n+2}$$Rn+2, isotropic surfaces in $$S^4$$S4 and homogeneous Willmore tori via the loop group method.

^{[27]}Further, it was shown that EDM resulted in shallow tensile residual stresses in the surface and an isotropic surface texture with higher surface roughness.

^{[28]}We consider a variant of Gamow's liquid drop model with an anisotropic surface energy.

^{[29]}This study investigated the sliding behaviour of water droplets depending on an anisotropic surface pattern.

^{[30]}The structural control-based manipulation of the anisotropic surface plasmon resonance (SPR) property of gold nanorod (AuNR) is significant and has wide application value in many fields.

^{[31]}The results show that more linear surface topographies and dense morphology have a large potential in the regeneration of musculoskeletal tissue, while nonpatterned scaffolds or more anisotropic surface microstructures present largest potential to promote.

^{[32]}Here, we introduce a special type of such structures formed by two penetrable metasurfaces which have complementary isotropic surface impedances.

^{[33]}Owing the fact that the ground surfaces were weaker anisotropic surfaces, the second order statistical evaluation methods are used to extract the changes in the image texture due to the variation in surface roughness of the component.

^{[34]}A sheet of graphene under magnetic bias attains anisotropic surface conductivity, opening the door for realizing compact devices such as Faraday rotators, isolators and circulators.

^{[35]}The results include the discussion of anisotropic surface structures induced by diamond turning or polishing as well as the analysis of the influence of a protection layer.

^{[36]}The results show that light couples into both localized and propagating plasmon modes associated with anisotropic surface protrusions and the gaps between such protrusions.

^{[37]}We introduce a novel algorithm for isotropic surface remeshing which progressively eliminates obtuse triangles and improves small angles.

^{[38]}For example, when a small hard sphere in adhesive contact with a soft silicone gel is slowly retracted from its rest position, it was found that the retraction force versus displacement relation cannot be explained either by the Johnson-Kendall-Roberts (JKR) theory or a recent indentation theory based on an isotropic surface stress that is independent of surface strain.

^{[39]}Here, the origin of microscopic liquid residue is studied by investigating directional drop self-transport on anisotropic surfaces made of radially arranged omniphobic microstripes.

^{[40]}Finally, we give an example to compute isophote curves on isotropic surfaces of revolution.

^{[41]}The bidirectional reflectance distribution function (BRDF) data are essential for analyzing and modeling the material appearance of anisotropic surfaces.

^{[42]}Friction anisotropic surfaces with oriented micro-/nanostructures are widely spread in biological systems and play important roles for effective locomotion or transporting items.

^{[43]}Pronounced anisotropic surface roughening and surface melting is observed and compared with earlier studies of four metals with different crystal structures illustrating significant differences in disordering behavior.

^{[44]}Canonical problems for evaluating grid convergence of the numerical method and validation tests for stability of a growing nucleus in the presence of isotropic surface tension are presented.

^{[45]}Within the framework of highly anisotropic surface elasticity model we discuss the propagation of new type of surface waves that are anti-plane surface waves.

^{[46]}The anisotropic surface chemistry of different surfaces on the crystal influences its physical and chemical properties, such as solubility, tableting, etc.

^{[47]}Illuminated n-GaP, in contrast, showed initially high corrosion faradaic efficiency which decreased during operation and was attributed to the phenomena of anisotropic surface etching and micropore formation during the beginning stages of photoanodic operation.

^{[48]}Incorporating anisotropic surface charges on atomically precise gold nanoclusters (Au NCs) led to a strong absorption in the near-infrared region and could enable the formation of self-assembled Au NCs exhibiting an intense absorption band at ∼1000 nm.

^{[49]}The design of submicron- and nanoscaled isotropic or anisotropic surfaces leads to the activation of extraordinary surface and optical properties.

^{[50]}

## isotropic surface energy

We revisit the crack kinking problem in materials with strongly anisotropic surface energies by using a variational phase-field model.^{[1]}First, based on the thermodynamic variation and smooth vector-field perturbation method, we rigorously derive a sharp-interface model with weakly anisotropic surface energies, and this model describes the interface evolution which occurs through surface diffusion flow and contact line migration.

^{[2]}A novel phase field material point method is introduced for robust simulation of dynamic fracture in elastic media considering the most general case of anisotropic surface energy.

^{[3]}The described limited crystal phase nucleation, the circumstances of which are unique in the realm of small-molecule organic semiconductors, is explained in terms of a two step nucleation mechanism while taking into account the anisotropic surface energy of crystal grains.

^{[4]}To this end, a novel description of the local thermodynamic equilibrium of frozen water (ice) and a mixture of water vapour and an inert gas is derived, where anisotropic surface energy densities can be considered.

^{[5]}Given an elliptic integrand of class $ \mathscr{C}^{3} $, we prove that finite unions of disjoint open Wulff shapes with equal radii are the only volume-constrained critical points of the anisotropic surface energy among all sets with finite perimeter and reduced boundary almost equal to its closure.

^{[6]}We consider a variant of Gamow's liquid drop model with an anisotropic surface energy.

^{[7]}The preferred growth direction of the α-Mg dendrite is found to be dependent on the magnitude of the anisotropic surface energy and the difference of crystallographic anisotropy between the matrix Mg and the additional solute X.

^{[8]}We study a variational problem for hypersurfaces in the Euclidean space with an anisotropic surface energy.

^{[9]}The ∑13a grain boundaries are likely to form in the cemented carbides with anisotropic surface energy by coalescence of WC platelets with large shape aspect ratio.

^{[10]}(2015), ‘Phase-field modeling and simulation of fracture in brittle materials with strongly anisotropic surface energy’.

^{[11]}Furthermore, anisotropic surface energies of different oriented crystals lead to anisotropy of nitrogen adsorption process.

^{[12]}

## isotropic surface impedance

In this Letter, to the best of our knowledge, a novel anisotropic surface impedance holographic metasurface antenna for generating orbital angular momentum (OAM) is developed through design, fabrication, and measurement at radio frequencies.^{[1]}Here, we introduce a special type of such structures formed by two penetrable metasurfaces which have complementary isotropic surface impedances.

^{[2]}The synthesized anisotropic surface impedance provides transformation of the incident fields of the sources into a plane wave scattered in a prescribed direction with a given polarization.

^{[3]}

## isotropic surface tension

The dynamics of interfacial evolution of a particle in the binary alloy melt affected by anisotropic surface tension is studied by means of the multiple variable expansion method.^{[1]}This paper studies the effect of anisotropic surface tension on the morphological stability of deep cellular crystal in directional solidification by using the matched asymptotic expansion method a.

^{[2]}Canonical problems for evaluating grid convergence of the numerical method and validation tests for stability of a growing nucleus in the presence of isotropic surface tension are presented.

^{[3]}

## isotropic surface remeshing

We present a novel anisotropic surface remeshing method that can efficiently eliminate obtuse angles.^{[1]}We introduce a novel algorithm for isotropic surface remeshing which progressively eliminates obtuse triangles and improves small angles.

^{[2]}

## isotropic surface pattern

An anisotropic surface pattern named polar microstructure is proposed to provide the global map for the matching.^{[1]}This study investigated the sliding behaviour of water droplets depending on an anisotropic surface pattern.

^{[2]}

## isotropic surface reactivity

The broken bond density on the exposed surfaces of fluorite was calculated, enabling the anisotropic surface reactivity of fluorite such as surface energy, surface relaxation, dissolution, wettability, surface charge and adsorbability to be revisited.^{[1]}We find that an anisotropic surface reactivity alone is enough to give rise to active linear motion of a JP, whereas active rotation only occurs if the JP is not axisymmetric.

^{[2]}

## isotropic surface plasmon

The structural control-based manipulation of the anisotropic surface plasmon resonance (SPR) property of gold nanorod (AuNR) is significant and has wide application value in many fields.^{[1]}In this configuration, the anisotropic surface plasmon polaritons (SPPs) supported by the GS are excited and provide a new channel for the near-field energy transport.

^{[2]}