Isoconcentration Surfaces Research Articles

Effect of Cu addition on the precipitation behavior of M2C carbides in maraging steel

In this study, two Ferrium S53 steels with and without 1.5% Cu were prepared. The microscopic morphology and structure of these steels during aging treatment were analyzed by transmission electron microscopy (TEM). The average equivalent spherical radius (Rv), volume fraction (ϕv), and number density (Nv) of Cu-rich precipitates and M2C carbides during aging treatment were analyzed by atom probe tomography (APT), and the spatial distribution, interfacial elemental segregation, and one-dimensional concentration profiles of the two types precipitates were also characterized. Based on the distribution of elements, 5%Cu and 9% (C + Mo) isoconcentration surfaces were selected to obtain Cu-rich precipitates and M2C carbides, respectively. The results demonstrate that the Cu-rich precipitates formed before the M2C carbides. The segregation of Cr and Mo elements at the phase interface due to the nucleation and growth of Cu-rich precipitates can promote the formation of the M2C carbides. The two types of precipitates display adjacent distribution relationship in steel. The number density of M2C carbides in steel contain 1.5%Cu can be greatly increased when the Cu-rich precipitates serving as nucleation particles. The formation of needle-like M2C carbides inhibited the coarsening of Cu-rich precipitates. More dispersive and fine M2C carbides were obtained in the steel.

Three-Dimensional Reconstruction of Interface Roughness and Alloy Disorder in Ge/GeSi Asymmetric Coupled Quantum Wells Using Electron Tomography.

Interfaces play an essential role in the performance of ever-shrinking semiconductor devices, making comprehensive determination of their three-dimensional (3D) structural properties increasingly important. This becomes even more relevant in compositional interfaces, as is the case for Ge/GeSi heterostructures, where chemical intermixing is pronounced in addition to their morphology. We use the electron tomography method to reconstruct buried interfaces and layers of asymmetric coupled Ge/Ge0.8Si0.2 multiquantum wells, which are considered a potential building block in THz quantum cascade lasers. The three-dimensional reconstruction is based on a series of high-angle annular dark-field scanning transmission electron microscopy images. It allows chemically sensitive investigation of a relatively large interfacial area of about (80 × 80) nm2 with subnanometer resolution as well as the analysis of several interfaces within the multiquantum well stack. Representing the interfaces as iso-concentration surfaces in the tomogram and converting them to topographic height maps allows the determination of their morphological roughness as well as layer thicknesses, reflecting low variations in either case. Simulation of the reconstructed tomogram intensities using a sigmoidal function provides in-plane-resolved maps of the chemical interface widths showing a relatively large spatial variation. The more detailed analysis of the intermixed region using thin slices from the reconstruction and additional iso-concentration surfaces provides an accurate picture of the chemical disorder of the alloy at the interface. Our comprehensive three-dimensional image of Ge/Ge0.8Si0.2 interfaces reveals that in the case of morphologically very smooth interfaces─depending on the scale considered─the interface alloy disorder itself determines the overall characteristics, a result that is fundamental for highly miscible material systems.

Application of electron tomography for comprehensive determination of III-V interface properties

We present an electron tomography method for the comprehensive characterization of buried III-V semiconductor interfaces that is based on chemical-sensitive high-angle annular dark-field scanning transmission electron microscopy. For this purpose, an (Al,Ga)As/GaAs multi-layer system grown by molecular beam epitaxy is used as a case study. Isoconcentration surfaces are exploited to obtain topographic height maps of 120 nm × 120 nm area, revealing the interface morphology. By applying the height-height correlation function, we are able to determine important interface properties like root mean square roughness and lateral correlation length of various interfaces of the (Al,Ga)As/GaAs system characterized by different Al concentrations. Height-difference maps based on isosurfaces corresponding to 30% and 70% of the total compositional difference at the interfaces are used to create topographic maps of the interface width and to calculate an average interface width. This methodology proves differences in the properties of direct and inverted interfaces and allows the observation of interfacial anisotropies.

Atomic scale identification of nano-sized precipitates of Ta/Ti-added RAFM steel and its superior creep strength

The influence of 0.015 wt% Ti addition on the formation of MX precipitates and the creep resistances of a reduced activation ferritic/martensitic (RAFM) steel has been studied. 0.1 wt% Ta was also added as same as conventional RAFM steels. Transmission electron micrographs taken from extraction replicas indicated that the area fraction of MX particles in Ta/Ti-added RAFM steel was 2.3 times higher than that in the reference steel. Atom probe tomography was employed to identify the types of MX precipitates and their interactive distribution. By using isoconcentration surfaces with different elemental concentration values, it was newly found that there are three types of MX precipitates, i.e. (Ta,V)-rich MX, Ti-rich MX and W-rich MX in the Ta/Ti-added RAFM steel. They appeared to be distributed independently in the matrix. However, in some cases, smaller sized (Ti or W)-rich MX particles were in contact with large (Ta,V)-rich MX particle. The creep rupture life of the Ta/Ti-added RAFM steel was significantly improved, as compared with the reference steel. The enhanced creep resistance can be rationalized in terms of a high density of dislocations, which were produced by a strong interaction with a higher fraction of the nano-sized MX particles within laths.

3D Atomic Mapping of Interfacial Roughness and Its Spatial Correlation Length in Sub-10 nm Superlattices.

The interfacial abruptness and uniformity in heterostructures are critical to control their electronic and optical properties. With this perspective, this work demonstrates the three-dimensional (3D) atomic-level mapping of the roughness and uniformity of buried epitaxial interfaces in Si/SiGe superlattices with a layer thickness in the 1.5-7.5 nm range. Herein, 3D atom-by-atom maps were acquired and processed to generate isoconcentration surfaces highlighting local fluctuations in content at each interface. These generated surfaces were subsequently utilized to map the interfacial roughness and its spatial correlation length. The analysis revealed that the root-mean-squared roughness of the buried interfaces in the investigated superlattices is sensitive to the growth temperature with a value varying from 0.17 ± 0.02 to 0.26 ± 0.03 nm in the temperature range of 500-650 °C. The estimated horizontal correlation lengths were found to be 8.11 ± 0.5 nm at 650 °C and 10.09 ± 0.6 nm at 500 °C. Additionally, reducing the growth temperature was found to improve the interfacial abruptness, with a 30% smaller interfacial width is obtained at 500 °C. This behavior is attributed to the thermally activated atomic exchange at the surface during the heteroepitaxy. Finally, by testing different optical models with increasing levels of interfacial complexity, it is demonstrated that the observed atomic-level roughening at the interface must be accounted for to accurately describe the optical response of Si/SiGe heterostructures.

Phase Segmentation in Atom-Probe Tomography Using Deep Learning-Based Edge Detection

Atom-probe tomography (APT) facilitates nano- and atomic-scale characterization and analysis of microstructural features. Specifically, APT is well suited to study the interfacial properties of granular or heterophase systems. Traditionally, the identification of the interface between, for precipitate and matrix phases, in APT data has been obtained either by extracting iso-concentration surfaces based on a user-supplied concentration value or by manually perturbing the concentration value until the iso-concentration surface qualitatively matches the interface. These approaches are subjective, not scalable, and may lead to inconsistencies due to local composition inhomogeneities. We introduce a digital image segmentation approach based on deep neural networks that transfer learned knowledge from natural images to automatically segment the data obtained from APT into different phases. This approach not only provides an efficient way to segment the data and extract interfacial properties but does so without the need for expensive interface labeling for training the segmentation model. We consider here a system with a precipitate phase in a matrix and with three different interface modalities—layered, isolated, and interconnected—that are obtained for different relative geometries of the precipitate phase. We demonstrate the accuracy of our segmentation approach through qualitative visualization of the interfaces, as well as through quantitative comparisons with proximity histograms obtained by using more traditional approaches.

The Influence of Isoconcentration Surface Selection in Quantitative Outputs from Proximity Histograms.

Isoconcentration surfaces are commonly used to delineate phases in atom probe datasets. These surfaces then provide the spatial and compositional reference for proximity histograms, the number density of particles, and the volume fraction of particles within a multiphase system. This paper discusses the influence of the isoconcentration surface selection value on these quantitative outputs, using a simple oxide dispersive strengthened alloy, Fe91Ni8Zr1, as the case system. Zirconium reacted with intrinsic oxygen impurities in a consolidated ball-milled powder to precipitate nanoscale zirconia particles. The zirconia particles were identified by varying the Zr-isoconcentration values as well as by the maximum separation data mining method. The associated outputs mentioned above are elaborated upon in reference to the variation in this Zr isosurface value. Considering the dataset as a whole, a 10.5 at.% Zr isosurface provided a compositional inflection point for Zr between the particles and matrix on the proximity histogram; however, this value was unable to delineate all of the secondary oxide particles identified using the maximum separation method. Consequently, variations in the number density and volume fraction were observed as the Zr isovalue was changed to capture these particles resulting in a loss of the compositional accuracy. This highlighted the need for particle-by-particle analysis.

Structure and Optoelectronic Properties of Atomically Random Sn-Rich Gesn Semiconductors

The monolithic integration of group IV optoelectronic devices on silicon platform is a long-sought-for paradigm holding the key to a wealth of opportunities in ultrafast data transfer, low-power electronics, energy conversion, and sensing, to name a few. This all-group IV integrated photonics has recently gained a renewed interest with the re-emergence of direct band gap germanium-tin (GeSn) semiconductors. GeSn layers with Sn contents above 10% are commonly grown on a Ge virtual substrate (VS) on Si wafers. In recent years, digermane (Ge2H6) has been the canonical precursor to accomplish the growth of various Sn-rich GeSn layers and heterostructures.[1,2] However, Monogermane (GeH4) has recently been proposed as an alternative precursor for large-scale, CMOS-compatible growth of GeSn semiconductor lasers.[3-4] Here, we present the the growth and properties of metastable Sn-rich GeSn layers from the wafer-level down to the atomic scale.[5] The chemical vapor deposition (CVD) growth of GeSn multi-layer structures with Sn content in the 7-18% range was performed on a ~600 nm-thick Ge layer on 4-inch silicon wafers using Monogermane (GeH4) and tin-tetrachloride (SnCl4) precursors. The Sn incorporation is controlled by the change in temperature, ranging from 320°C, 300 °C, and 280 °C for the bottom (BL), middle (ML), and top (TL) GeSn layers, respectively. This leads to a BL/ML/TL stacking in Fig. 1a with Sn contents of 8.2%/11-13.5%/18%. Using transmission electron microscopy (TEM) measurements a TL/ML/BL GeSn layers stacking with a total thickness of ~380 nm grown on a ~600 nm-thick Ge-VS is observed. No threading dislocations propagating toward the GeSn TL are observed, while misfit and edge dislocations are confined at the Ge-GeSn interface and within the first 60-70 nm of the GeSn stacking. Herein, the lower Sn content BL and ML seem to minimize or even suppress the propagation of dislocations towards the defect-free 18% Sn TL. Atom probe tomography (APT) measurements are performed to establish an atomistic-level understanding of the evolution of Sn content across the grown layer. The APT profile for the Sn incorporation in Fig. 1b reveals the presence of abrupt interfaces between the different GeSn layers, with the highest incorporation of 18% in the TL. To provide details regarding the short-range atomic distribution on the scale of few lattice constants a series of four different statistical APT analyses will be discussed: frequency distribution analysis; partial radial distribution function; nearest-neighbor analysis; and iso-concentration surfaces.[5-6] The APT analysis demonstrates the growth of a GeSn random alloy, with a perfect random distribution of Sn atoms within the grown layers, and without the presence of short-range ordering effects and Sn precipitates. Interestingly, the defect-free highest incorporation of Sn is achieved at a growth rate (~1.3 nm/min) that is one order of magnitude lower than other studies where either GeH4 or Ge2H6 precursors were used.[1-3] It is clear that the use of GeH4 allows for additional set of growth parameters were a high incorporation of Sn still be achieved with a drastic decrease reduction (one order of magnitude) in the growth rate. The presence of multiple experimental growth conditions with extremely different growth rates suggests that the growth rate is a marginal factor in the Sn incorporation process, whereas the strain relaxation using lower Sn content buffered layers (BL/ML) will be discussed as a key factor in enhancing the incorporation of Sn. We therefore highlight the importance of reducing strain at the growth surface while keeping a low growth temperature, which can enhance the incorporation of Sn. This behavior will facilitate the growth of GeSn multilayer structures enabling more advanced strain and band structure engineering. The sharp interfaces across GeSn layers pave the way for achieving a high degree of control of the growth of GeSn-based multi-quantum wells heterostructures for mid-IR sources and detectors. Furthermore, by combining theoretical calculations and experiments, the presentation will also discuss the influence of structure and composition on the optoelectronic properties of the as-grown GeSn layers. Acknowledgements The authors thank J. Bouchard for the technical support, and NSERC Canada (Discovery, SPG, and CRD Grants), Canada Research Chair, Canada Foundation for Innovation, Mitacs, FQRNT, and MRIF Québec for support.

Atomistic and Optical Properties of Group IV Ultrathin Superlattices

Engineering atomically abrupt interfaces between multi-layered thin films of different material have yielded some significant fundamental and technological breakthroughs in the past. For example, the Nobel Prize winning integral and quantum Hall effect owes its discovery to abrupt semiconductor interfaces grown by molecular beam epitaxy. On the other hand, the functionalization of many modern-day semiconductor devices depend critically on the quality of interface of the various thin films, which constitute the device. Silicon (Si)/Germanium (Ge) and Si/SiGe based materials have emerged in the recent years as an integral part to the Si semiconductor industry, promising a wide range of devices ranging from Fin-field effect transistors, quantum cascade lasers, quantum dots, near and mid-infrared photodetectors, resonant tunnel diodes, to name a few.1–3 To this end, a continuous feedback from post-growth characterization techniques elucidating the intermixing of atoms at the interface, the interfacial width and roughness, the defect density at the interfaces as a function of the growth conditions and integration parameters is therefore crucial for optimizing the ultimate performance of these devices. Not surprising that the semiconductor industry is directing a significant amount of its resources and efforts towards enriching our current understanding of the buried interfaces of these materials. In this work, we use a combination of scanning transmission electron microscopy (STEM) and atom probe tomography (APT) to present an atomistic understanding of the nature of interfaces in Si/Ge and Si/SiGe superlattices, grown on 300 mm Si wafers using chemical vapor deposition. The APT investigations were done using the state-of-the-art LEAP 5000 providing a supreme mass sensitivity and a detection efficiency of 80%, a 30% improvement compared to its predecessor. Figure 1(a) shows the STEM-APT of the last six bi-layers of a sixteen period Si/SiGe superlattice. Figure 1(b) shows the one-dimensional (1D) concentration profile obtained from the 3D APT reconstruction of the sample along the growth (analysis) direction. While the STEM data reveals the average Si and SiGe layer thickness to be about 1.3 nm and 2.2 nm respectively, the concentration profile from the APT data reveals the Ge content within the SiGe layers to be 25.0 +/- 0.5 at.%. The average interfacial width, evaluated as the separation between the 10.0 at.% and 90.0 at.% Ge concentration points, revealed the Si to SiGe transition to be about 1.4 +/- 0.3 nm wide while the width of the SiGe to Si transition to be narrower than 1.0 nm. The possible role of Ge diffusion on the observed interfacial thicknesses shall be discussed in details in this work. Figure 1(c) shows the Si/SiGe hetero-interfaces drawn as isoconcentration surfaces at 12.0 at.% Ge concentration. The work shall also highlight the results obtained from the STEM-APT measurements on superlattices of different period lengths, layer thicknesses and growth conditions from that shown in Figure 1. Alongside the structural aspects, the vibrational properties of these superlattices as obtained from Raman spectroscopic measurements and optical properties as obtained from the ellipsometry and absorption measurements shall be highlighted. Such superlattices holds tremendous prospect for Si semiconductor industry wherein a magic sequence of the number of monolayers within each period can render the whole superlattice to be direct gap4 for optoelectronic applications or an array of ultrathin free-standing nanowires could be fabricated by selective etching of the Ge layers in a Si/Ge superlattice for nanoelectronic applications. (1) Tan, K.-M.; Liow, T.-Y.; Lee, R. T. P.; Hoe, K. M.; Tung, C.-H.; Balasubramanian, N.; Samudra, G. S.; Yeo, Y.-C. IEEE Electron Device Lett. 2007, 28 (10), 905–908. (2) Dehlinger, G.; Diehl, L.; Gennser, U.; Sigg, H.; Faist, J.; Ensslin, K.; Grutzmacher, D.; Muller, E. Science 2000, 290 (5500), 2277–2280. (3) Maune, B. M.; Borselli, M. G.; Huang, B.; Ladd, T. D.; Deelman, P. W.; Holabird, K. S.; Kiselev, A. A.; Alvarado-Rodriguez, I.; Ross, R. S.; Schmitz, A. E.; Sokolich, M.; Watson, C. A.; Gyure, M. F.; Hunter, A. T. Nature 2012, 481 (7381), 344–347. (4) d’Avezac, M.; Luo, J.-W.; Chanier, T.; Zunger, A. Phys. Rev. Lett. 2012, 108 (2), 27401. Figure 1

Core-Shell Structure of Intermediate Precipitates in a Nb-Based Z-Phase Strengthened 12% Cr Steel.

In creep resistant Z-phase strengthened 12% Cr steels, MX (M=Nb, Ta, or V, and X=C and/or N) to Z-phase (CrMN, M=Ta, Nb, or V) transformation plays an important role in achieving a fine distribution of Z-phase precipitates for creep strengthening. Atom probe tomography was employed to investigate the phase transformation in a Nb-based Z-phase strengthened trial steel. Using iso-concentration surfaces with different concentration values, and subtracting the matrix contribution enabled us to reveal the core-shell structure of the transient precipitates between MX and Z-phase. It was shown that Z-phase forms by diffusion of Cr into NbN upon ageing, and Z-phase has a composition corresponding to Cr1+x Nb1-x N with x=0.08.

Correlations and scaling properties of nonequilibrium fluctuations in liquid mixtures.

Diffusion in liquids is accompanied by nonequilibrium concentration fluctuations spanning all the length scales comprised between the microscopic scale a and the macroscopic size of the system, L. Up to now, theoretical and experimental investigations of nonequilibrium fluctuations have focused mostly on determining their mean-square amplitude as a function of the wave vector. In this work, we investigate the local properties of nonequilibrium fluctuations arising during a stationary diffusion process occurring in a binary liquid mixture in the presence of a uniform concentration gradient, ∇c_{0}. We characterize the fluctuations by evaluating statistical features of the system, including the mean-square amplitude of fluctuations and the corrugation of the isoconcentration surfaces; we show that they depend on a single mesoscopic length scale l=sqrt[aL] representing the geometric average between the microscopic and macroscopic length scales. We find that the amplitude of the fluctuations is very small in practical cases and vanishes when the macroscopic length scale increases. The isoconcentration surfaces, or fronts of diffusion, have a self-affine structure with corrugation exponent H=1/2. Ideally, the local fractal dimension of the fronts of diffusion would be D_{l}=d-H, where d is the dimensionality of the space, while the global fractal dimension would be D_{g}=d-1. The transition between the local and global regimes occurs at a crossover length scale of the order of the microscopic length scale a. Therefore, notwithstanding the fact that the fronts of diffusion are corrugated, they appear flat at all the length scales probed by experiments, and they do not exhibit a fractal structure.

Creating Isoconcentration Surfaces in Low-Chemical-Partitioning, High Solute-Containing Alloys

Creating Isoconcentration Surfaces in Low-Chemical-Partitioning, High Solute-Containing Alloys

Extracting features buried within high density atom probe point cloud data through simplicial homology

Feature extraction from Atom Probe Tomography (APT) data is usually performed by repeatedly delineating iso-concentration surfaces of a chemical component of the sample material at different values of concentration threshold, until the user visually determines a satisfactory result in line with prior knowledge. However, this approach allows for important features, buried within the sample, to be visually obscured by the high density and volume (~107 atoms) of APT data. This work provides a data driven methodology to objectively determine the appropriate concentration threshold for classifying different phases, such as precipitates, by mapping the topology of the APT data set using a concept from algebraic topology termed persistent simplicial homology. A case study of Sc precipitates in an Al–Mg–Sc alloy is presented demonstrating the power of this technique to capture features, such as precise demarcation of Sc clusters and Al segregation at the cluster boundaries, not easily available by routine visual adjustment.

A procedure to create isoconcentration surfaces in low-chemical-partitioning, high-solute alloys

A proximity histogram or proxigram is the prevailing technique of calculating 3D composition profiles of a second phase in atom probe tomography. The second phase in the reconstruction is delineated by creating an isoconcentration surface, i.e. the precipitate–matrix interface. The 3D composition profile is then calculated with respect to this user-defined isoconcentration surface. Hence, the selection of the correct isoconcentration surface is critical. In general, the preliminary selection of an isoconcentration value is guided by the visual observation of a chemically partitioned second phase. However, in low-chemical -partitioning systems, such a visual guide is absent. The lack of a priori composition information of the precipitate phase may further confound the issue. This paper presents a methodology of selecting an appropriate elemental species and subsequently obtaining an isoconcentration value to create an accurate isoconcentration surface that will act as the precipitate–matrix interface. We use the H-phase precipitate in the Ni–Ti–Hf shape memory alloy as our case study to illustrate the procedure.

Radius dependence of solute concentration estimates of simulated ultrafine precipitates

Estimates of the radii and solute concentrations of simulated microstructures containing ultrafine spherical precipitates were determined from isoconcentration surfaces and proximity histograms. The accuracy of the estimates of the solute concentrations and the radii of precipitates was found to depend on the size of precipitates. Optimized parameters for analyzing 0.5- to 2-nm-radius precipitates are proposed. The solute content of 0.5-nm-radius precipitates was not estimated correctly by this method. The accuracy of the estimates of the solute concentration and the radius of precipitates were primarily influenced by the solute concentrations of the precipitates. The ranges of error of the solute concentration in the precipitates, which are associated with the analytical limitations of the ultrafine precipitates, were determined, and the results indicated a limitation of the estimates.

Defect analysis by statistical fitting to 3D atomicmaps

In this article we present a statistical fitting method for evaluation of atomic reconstructions which does not require a coarse-graining step. The fitting compares different models of chemical structure in their capability to explain the measured data set by a least square type merit function. Only preliminary qualitative assumptions about the possible chemical structure are required, while accurate quantitative parameters of the chosen model are delivered by fitting. The technique is particularly useful for singular defect structures with very high composition gradients, for which iso-concentration surfaces determined by coarse-graining become questionable or impossible. We demonstrate that particularly detailed information can be gained from triple junctions and grain boundaries.

Correlation and Morphology of Dopant Decomposition in Mn and Co Codoped Ge Epitaxial Films

Atom probe tomography (APT) was used to quantify inhomogeneities in the distribution of Mn and Co in doped epitaxial Ge thin films for which X-ray diffraction (XRD) studies indicate single phase material. The segregation of dopants into Co and Mn-rich regions with characteristic sizes was evident upon visual inspection of the APT reconstruction and a frequency distribution analysis of the concentration of Co, Mn, and Ge atoms verified that the composition fluctuations exceeded those of a random alloy. Isoconcentration surfaces were generated to establish the connectedness of regions enriched in Mn that have been proposed to enhance the Curie temperature in dilute magnetic semiconductors. The analysis demonstrates important contributions that APT can make to the understanding of magnetism in these materials.

Combined use of small-angle neutron scattering and atom probe tomography for the analysis of precipitates in a Fe-15 m% Co-25 m% Mo alloy

Atom probe tomography (APT) measurements were conducted to analyze nm-sized precipitates in a Fe-25 m% Co-15 m% Mo model alloy. Calculation of isoconcentration surfaces and application of cluster-search algorithms yielded the chemical compositions of matrix and precipitates. Small-angle neutron scattering (SANS) was performed in order to calculate the ratio of magnetic to nuclear scattering. This so-called A-ratio was compared with values calculated from the chemical information derived from the atom probe data. For this, the magnetism of the precipitates has to be taken into account. Ab-initio studies have been carried out to determine the magnetic moment of the precipitate phase. The results show that the extremely fine precipitates are ferromagnetic and exhibit near-equilibrium composition. In the very early stage of decomposition, however, the composition is different from near-equilibrium. The influences of the magnetism of the precipitates as well as of artifacts from atom probe measurements on the obtained results are discussed.

Atom probe tomography

This introductory tutorial describes the technique of atom probe tomography for materials characterization at the atomic level. The evolution of the technique from the initial atom probe field ion microscope to today's state-of-the-art three dimensional atom probe is outlined. An introduction is presented on the basic physics behind the technique, the operation of the instrument, and the reconstruction of the three-dimensional data. The common methods for analyzing the three-dimensional atom probe data, including atom maps, isoconcentration surfaces, proximity histograms, maximum separation methods, and concentration frequency distributions, are described.

ON ISOCONCENTRATION SURFACES OF THREE-DIMENSIONAL TURING PATTERNS

We consider three-dimensional Turing patterns and their isoconcentration surfaces corresponding to the equilibrium concentration of the reaction kinetics. We call these surfaces equilibrium concentration surfaces (EC surfaces). They are the interfaces between the regions of "high" and "low" concentrations in Turing patterns. We give alternate characterizations of EC surfaces by means of two variational principles, one of them being that they are optimal for diffusive transport. Several examples of EC surfaces are considered. Remarkably, they are often very well approximated by certain minimal surfaces. We give a dynamical explanation for the emergence of Scherk's surface in certain cases, a structure that has been observed numerically previously in [De Wit et al., 1997].