Ta Alloys Research Articles

Temperature dependence of electroplastic effect on reducing the ultimate stress in Ti–6Al–2Zr–1Mo–1V alloy during tension

To investigate the electroplastic effect of Ti–6Al–2Zr–1Mo–1V alloy (TA15 alloy) at different temperatures, the uniaxial tension tests assisted with the pulsed current and temperature matched tension tests without current were performed at the temperature ranging from 200 °C to 800 °C in this study. The results demonstrated that the electroplastic effect of TA15 alloy became appreciable as the temperature approached 600 °C, namely that the pulsed current contributed about 12% reduction in ultimate stress with respect to that of temperature matched test when the temperature was higher than 600 °C. The microstructure observation showed that the pulsed current reduced the ultimate stress by enhancing dislocation recovery and α→β phase transformation. These electroplastic effects for reducing the ultimate stress were related on temperature. When the temperature was higher than 700 °C, the effect of pulsed current on ultimate stress drop changed from promoting dislocation motion to accelerating the nucleation and growth of softer β phase by accelerating the diffusion of Mo, V elements enriched in the β phase. The electroplastic mechanism was considered as the ‘local heat spot’. The higher energy surrounding the defects could reduce a resistance to move an ion core for titanium, which may have a significant effect on dislocation motion and element diffusion promoted by pulsed current.

High-throughput measurements of interdiffusivities in fcc Co–Ni–Ta alloys at 1373K and 1473K

The composition-dependent interdiffusivities in the fcc Co–Ni–Ta alloys at 1373 K and 1473 K were deduced by using our newly developed numerical inverse scheme that combines two-dimensional (2D) simulations with diffusion triple experiments. This approach largely reduced the experimental efforts by analyzing only one single-phase diffusion triple at each temperature yet covering a much wider composition range than diffusion couples. The reliability of the high-throughput results was firstly verified by reproducing the experimental 2D composition profiles in each triple. In order to further validate the deduced interdiffusivities, several traditional one-dimensional (1D) diffusion couples were also devised, and the widely recognized Sauer-Freise method and Whittle-Green method were then employed to calculate the interdiffusivities for the binaries and at the intersection points within the ternary composition range, respectively. The interdiffusivities extracted from the inverse scheme agree well with those from the traditional approaches, which strongly proves the applicability of the present new scheme. Besides, the constructed main interdiffusivity planes at 1373 K and 1473 K provide an overview of the diffusion behavior in the fcc Co–Ni–Ta system and promote further kinetic studies.

Biomechanical Behavior of Narrow Dental Implants Made with Aluminum- and Vanadium-Free Alloys: A Finite Element Analysis.

Titanium (Ti) alloys used for narrow dental implants usually contain aluminum (Al) and vanadium (V) for improved resistance. However, those elements are linked to possible cytotoxic effects. Thus, this study evaluated the biomechanical behavior of narrow dental implants made with Al- and V-free Ti alloys by the finite element method. A virtual model of a partially edentulous maxilla received single implants (diameter: 2.7 and 2.9 mm; length: 10 mm) at the upper lateral incisor area, with respective abutments and porcelain-fused-to-metal crowns. Simulations were performed for each implant diameter and the following eight alloys (and elastic moduli): (1) Ti-6Al-4V (control; 110 GPa), (2) Ti-35Nb-5Sn-6Mo-3Zr (85 GPa), (3) Ti-13Nb-13Zr (77 GPa), (4) Ti-15Zr (113 GPa), (5) Ti-8Fe-5Ta (120 GPa), (6) Ti-26.88Fe-4Ta (175 GPa), (7) TNTZ-2Fe-0.4O (107 GPa), and (8) TNTZ-2Fe-0.7O (109 GPa). The implants received a labially directed total static load of 100 N at a 45° angle relative to their long axis. Parameters for analysis included the maximum and minimum principal stresses for bone, and von Mises equivalent stress for implants and abutments. Ti-26.88Fe-4Ta reaches the lowest maximum (57 MPa) and minimum (125 MPa) principal stress values, whereas Ti-35Nb-5Sn-6Mo-3Zr (183 MPa) and Ti-13Nb-13Zr (191 MPa) models result in the highest principal stresses (the 2.7 mm model surpasses the threshold for bone overload). Implant diameters affect von Mises stresses more than the constituent alloys. It can be concluded that the narrow implants made of the Ti-26.88Fe-4Ta alloy have the most favorable biomechanical behavior, mostly by mitigating stress on peri-implant bone.

Influence of Tantalum Addition on the Corrosion Passivation of Titanium-Zirconium Alloy in Simulated Body Fluid.

Ti-15%Zr alloy and Ti-15%Zr-2%Ta alloy were fabricated to be used in biomedical applications. The corrosion of these two alloys after being immersed in simulated body fluid for 1 h and 72 h was investigated. Different electrochemical methods, including polarization, impedance, and chronoamperometric current with time at 400 mV were employed. Also, the surface morphology and the compositions of its formed film were reported by the use of scanning electron microscope and energy dispersive X-ray. Based on the collected results, the presence of 2%Ta in the Ti-Zr alloy passivated its corrosion by minimizing its corrosion rate. The polarization curves revealed that adding Ta within the alloy increases the corrosion resistance as was confirmed by the impedance spectroscopy and current time data. The change of current versus time proved that the addition of Ta reduces the absolute current even at high anodic potential, 400 mV. The results of both electrochemical and spectroscopic methods indicated that pitting corrosion does not occur for both Ti-Zr and Ti-Zr-Ta alloys, even after their immersion in SBF solutions for 72 h.

Coexistence of memory and threshold resistive switching identified by combinatorial screening in niobium-tantalum system

Memristive devices exhibiting improved performance and memory characteristics are revealed by using a combinatorial screening approach. An Nb-Ta thin film library, with a total compositional spread ranging between 13 and 80 at.% Ta, is co-sputtered as the bottom electrode, while their anodic oxides served as active layers. Such metal–insulator-metal (MIM) specimens under investigation are finalized with patterned Pt top electrodes. Memristors based on mixed Nb2O5-Ta2O5 have been grouped according to their common switching characteristics in four different compositional zones. Reversible switching between threshold and non-volatile mode was observed for devices grown on Nb-39 at.% Ta alloy. The switching mode was dictated by applying different current compliances. Moreover, devices based on Nb-36 at.% Ta switch at minimum seven distinct resistive levels. Apart from the electrical measurements, the structure of samples and components distribution in cross-section geometry is studied via TEM techniques before and after electrical treatment. Electrolyte incorporation and the exact composition of the identified high-performance memristors (resulting from the combinatorial screening) are analyzed by HAXPES. The memristive devices based on the two singled-out Nb-Ta alloys show enhanced memory and electrical characteristics, thus being excellent candidates for neuromorphic applications.

Precipitate formation in the Fe-Al-Nb and Fe-Al-Ta systems

The composition of the metastable L21 Heusler phase in Fe2AlX with (X = Nb, Ta) alloys has been determined by atom probe tomography (APT). It was found that the composition of L21 is off-stoichiometric in both systems, however closer to the stoichiometric composition in the Fe-Al-Ta than in the Fe-Al-Nb(-B) alloy. L21 dissolves faster and therefore the formation of the stable C14 Laves phase proceeds quicker in the Fe-Al-Nb(-B) alloy. Doping with boron does not lead to the formation of borides and only a slight enrichment of boron in the Fe-Al matrix at the grain boundary, which is covered with C14 precipitates is observed.

Phase constituent and microstructure manipulations via annealing for enhancements of mechanical property and functionalities of Ti–Au–Cr–Ta biomedical shape memory alloys

Shape memory alloys (SMAs) are considered as one of the most important smart materials for the future biomedical applications in consideration of the rapid world population aging. The promising Ti–Au–Cr–Ta alloys, which are good in biocompatibility, excellent in X–ray contrast, and high in corrosion resistance, were synthesized and evaluated in this study for realizing new biomedical materials. The specimens with the specific chemical composition of Ti–4Au–5Cr–5Ta (at.%) were chosen as the target material based on our preliminary findings. The Ti–4Au–5Cr–5Ta alloys were thereafter subjected to annealing–treatments at various annealing time periods for the manipulations of the phase constituents and microstructures. The fundamental mechanical properties and functionalities, such as shape deformation/recovery behaviors in the loading/unloading processes, were investigated. The two optimized specimens, which were subjected to a 1073 K annealing–treatment for 1.8 ks and 3.6 ks, performed good mechanical properties (i.e., strength and elongation) and excellent shape recovery rate of 96 % and 97 %, respectively. Via the systematic fine–tuning strategies, the Ti–4Au–5Cr–5Ta alloy with appropriate heat–treatment surpassed all other Ti–Au–Cr–Ta alloys, which were study by our research group.

Interface Characteristics, Erosion Behavior, and Thermal Shock Resistance of Al–Ta Alloy Coatings Produced by Arc Spraying

The pure aluminum coatings of physical vapor deposition chambers used in current semiconductor processes offer insufficient protection against ion bombardment. In this study, the arc spraying process was used to produce Al–Ta alloy coatings to replace pure aluminum coatings. The microstructure, mechanical properties, and particle erosion properties of these coatings were investigated at 1% and 3% Ta content and compared with those of pure Al. The surface and surface morphologies of the three coatings were similar. However, Ta was uniformly dispersed to form a supersaturated solid solution in the 1Ta and 3Ta coatings. The tensile results indicated that the adhesion of the three coatings was similar. According to particle erosion results, cohesive strength and erosion resistance apparently increased. Moreover, the thermal shock effect caused by the extreme thermal cycle in these chambers causes coating degradation. So, this study compared the thermal shock properties of the Al–Ta alloy coatings with Y2O3 ceramic coating. The 1Ta and 3Ta coatings did not differ considerably from pure Al coating in terms of thermal shock resistance, and all three were higher than that of the Y2O3 coating. The aforementioned results indicate that the produced Al–Ta alloy coatings can offer superior chamber protection ability.

The thermo-mechanical behavior of NiTi-X shape memory alloys

A commercial NiTi alloy with the chemical composition of 54.5 Ni-45.5 Ti (mass %) was used as a standard material to be compared with Ti50Ni45Cu5, 48 Ni-37 Ti-15 Nb (mass. %) and 48 Ni- 37 Ti-15 Ta (mass. %). The experimental alloys were cast, annealed and hot rolled in order to render the material more compact and to obtain 0.5 mm-thick lamellas. Two types of specimens were then obtained. The former were fragments with a mass of less than 50 mg that were cut and cleaned in alcohol, in an ultrasonic bath, before being subjected to differential scanning calorimetry (DSC) analysis. The latter were 0.5 × 4 × 25 mm rectangular specimens that were prepared to be subjected to dynamic mechanical analyses (DMA) and optical analysis (OM). The DSC investigation program comprised a heating–cooling cycle with a rate of 10 K/min and was used in order to obtain the critical transformation temperatures of direct and reverse martensitic transformations. DMA measurements consisting in a heating cycle with a heating rate of 5 K/min, a frequency of 1 Hz, a maximum deformation of 20 µm and a maximum of 9 N dynamical applied force, were used in order to observe the mechanical behavior under dynamic conditions. DSC tests suggested that Cu and Ta alloying led to an increase of critical temperatures of reverse martensitic transformation while Nb alloying dropped these critical temperatures as low as 0 °C. In addition, Ta alloying significantly increased the transformation temperatures. All the DMA thermograms emphasized a maximum of internal friction that could be associated with the reverse martensitic transformation. Storage modulus exhibited either continuous or step-like increase during heating, suggesting the presence of a thermoelastic transformation.

Strengthening mechanism of lamellar-structured Ti–Ta alloys prepared by powder metallurgy

Titanium-tantalum alloys with lamellar microstructure were fabricated by cold isostatic pressing under 400 MPa and pressure-free sintering at 1600 °C. By such a low-cost powder metallurgy method, the Ti–Ta alloy exhibited the tensile yield strength of 1124 MPa at room temperature, which was twice that of the cast Ti–Ta alloys. The microstructure and mechanical properties were characterized in detail to elucidate the deformation behavior and the strengthening mechanism of the Ti–Ta alloys prepared by powder metallurgy. With the increasing Ta content, the width of the α laths decreased, and that of the β laths increased, leading to the precipitation of the acicular α in the β laths. Correspondingly, the yield and ultimate strength increased, while the strain to failure decreased. Lamellar structure strengthening was determined to be the dominating mechanism for the high strength of the Ti–Ta alloys prepared by powder metallurgy. After quantitatively evaluating the contributions of laths, solute atoms, prior β grain boundaries, and dislocations, a model was established to illustrate the yield strength of the Ti–Ta alloys. The calculated values agreed well with the measured strength values of the Ti–Ta alloys, indicating that this model is effective in forecasting the strength of the dual-phase Ti alloys with lamellar structure.

Tensile Deformation Behavior and Dynamic Recrystallization of TA15 Alloy over a Wide Temperature Range

TA15 is a near‐α titanium alloy that is used extensively in the manufacturing of high‐temperature structural components and has important applications in the aerospace field. Herein, the tensile deformation behavior of a TA15 alloy with equiaxed α grains is investigated over a wide temperature range of 25–900 °C. Based on the microstructural observations and tensile testing results, the deformation behavior of the TA15 alloy is classified into four stages: 1) deformation strengthening stage at 25–550 °C; 2) α‐phase softening stage at 600–650 °C; 3) phase transition softening stage at 700–800 °C; and 4) dynamic recrystallization stage at 850–900 °C. The α→β phase transition occurs predominantly between equiaxed α grains, and the increase in β‐phase content results in further enhancement of alloy plasticity. Dynamic recrystallization occurs first at the edge of the equiaxed α grains and gradually extends into the grain interior. The size of the dimples on the fracture surface increases gradually as the temperature increases, and a large shearing and splitting area is observed at 600 °C due to α‐phase softening.

Effect of scanning strategy on microstructure and mechanical properties of a biocompatible Ti–35Nb–7Zr–5Ta alloy processed by laser-powder bed fusion

The influence of scanning strategy (SS) on microstructure and mechanical properties of a Ti–35Nb–7Zr–5Ta alloy processed by laser-powder bed fusion (L-PBF) is investigated for the first time. Three SSs are considered: unidirectional-Y; bi-directional with 79° rotation (R79); and chessboard (CHB). The SSs affect the type and distribution of pores. The highest relative densities and more homogeneous distribution of pores are obtained with R79 and CHB scanning strategies, whereas aligned pores are formed in the unidirectional-Y. The SSs show direct influence on the crystallographic texture with unidirectional-Y strategy showing fiber texture. The R79 strategy results in a weak texture and the CHB scanning strategy forms a randomly oriented heterogeneous grain structure. The lowest Young modulus is obtained with the unidirectional-Y strategy, whereas the R79 strategy results in the highest yield strength. It is shown that the SSs may be used for tuning the microstructure of a beta-Ti alloy in L-PBF. Graphical abstract

Reactive co-sputtering of ternary Au–Ta–O films with tunable electrical resistivity

Heavy-metal-based films with high electrical resistivity are needed for hohlraums for magnetically assisted inertial confinement fusion. Here, we study ternary Au–Ta–O films deposited by reactive direct-current magnetron co-sputtering from elemental Au and Ta targets in an oxygen containing atmosphere. By varying the O content, the electrical resistivity of films can be tuned in a wide range of ∼100–40 000 μΩ cm. With increasing O content, a drastic increase in resistivity occurs at ∼45 at. % of O, separating regimes with two different dominant conduction mechanisms attributed to metallic conduction through the Au–Ta alloy matrix (for ≲45 at. % of O) and tunneling across insulating Ta2O5 layers separating conducting islands (for ≳45 at. % of O). Post-deposition annealing at 300 °C leads to the segregation of Au into ∼50-nm islands, sharply decreasing the resistivity for films with ≳45 at. % of O but not for the metal-like films with lower O content.

Exploration of homologous substitution element on phase ratio and high temperature properties in high Nb-containing TiAl alloy

To reveal homologous substitution and refined mechanisms, and improve mechanical properties, Ti42Al6Nb 2.5 CxTa alloys were prepared by arc melting. Element distribution, lamellar colony size, length-diameter radio of Ti2AlC phase, tensile properties, and the corresponding influence mechanism were studied. Results show that the relative contents of γ and Ti2AlC phase increase, the content of α2 phase decreases, when Ta content increases from 0 to 2.0 at%. The relative content of B2 phase increases among the lamellar colonies and the length-diameter ratio of Ti2AlC particle decreases. Lamellar colony size decreases from 24.9 to 10.6 µm. With the increase of Ta content, the supercooling at the front of the solid-liquid interface increases and the β phase is refined. The increase of the precipitated phase content leads to the increase of the number of α phase heterogeneous nucleation particles, which will refine the lamellar colony. Tensile results show that the strength increases from 238 to 513 MPa and strain increases from 2.7 % to 11.1 % of Ti42Al6Nb 2.5 C1.6Ta alloy when the tested temperature increases from 750 to 950 ℃. The high temperature tensile properties of the alloy with more Ta content are relatively higher. Ta forms large lattice distortion after solid solution into the matrix due to its large atomic size, and Ta makes more Nb dissolve into the collective to increase the effect of solid solution strengthening. Another reason is the refinement of microstructure.

Microstructure and mechanical properties of Ta–Ti alloy by plasma-activated sintering

Tantalum-based alloys have developed rapidly in the fields of aerospace, nuclear industry, weapons, and national defense heavy industries due to their high melting point, low ductility-brittle transition temperature, and well ductility. However, pure tantalum exhibits low room-temperature strength and intergranular fracture, which is attributable to its strong adsorption to interstitial oxygen during sintering. In this work, high-strength Ta–Ti alloy is achieved by introducing Ti to consume oxygen by forming TiO2 and inhabiting the solution of oxygen in Ta alloys. The Ta–Ti alloys mainly consist of dense Ta matrix and dispersed TiO2 second phase. With the addition of 2 wt% Ti, the Ta–Ti alloy with high densification of 99.2% not only exhibits high ultimate bending strength of 1345 MPa from dispersion strengthening by TiO2 oxides, but also has good plasticity of 6% with a transgranular fracture and numerous dimple. The present work may facilitate the development and industrial application of Ta-based alloys.

Microstructural and texture evolution investigation of laser melting deposited TA15 alloy at 500°C using in-situ EBSD tensile test

The deformation behavior in polycrystalline Laser melting deposited (LMD) TA15 alloy was investigated at a temperature of 500°C using advanced in-situ Secondary electron microscopy (SEM) tensile setup coupled with Electron backscatter diffraction (EBSD). The grains morphology, crystal orientation, preferred pole texture, activated slip traces, grain boundaries evolution, and strain-induced misorientation were systematically analyzed during tensile straining. It was determined that the diverse α-grains (crystal orientation and morphology) revealed inhomogeneous deformation under tensile loading. The overall strain was allocated by grain rotation and substructure formation. The preferred oriented (0001) pole texture maximum was increased during initial strain level. In contrast, at higher strain, the texture intensity was decreased due to the formation and relocation of the newly formed substructures. The fraction of low angle grain boundaries (LAGBs 2–10°) was increased continually with increasing strain. Strain-induced dislocations within larger grains formed new LAGBs, and the increasing fraction of LAGBs was validated by Kernel average misorientation (KAM) map. The prismatic slip mode was identified as the primary active slip system at the initial deformation stage. The fraction of basal and pyramidal <a> slips increased as the strain increased, describing the opposite trend to prismatic slip. The grains rotation and strain accumulation at the grain boundaries accommodated the plastic deformation, ultimately causing boundary sliding.

Solubility and mechanical properties of hydrogen / carbon in Mo–Ta alloy

The solubility, diffusivity, electronic and mechanical properties of Hydrogen (H) and Carbon(C) impurity atoms in Mo–Ta alloy were studied by first-principles calculation. The solubility energy results of C/H in Mo–Ta alloy show that H tends to form in tetrahedral interstitial site (TIS), in contrast to C which tends to octahedral interstitial site (OIS). The diffusion barrier of H/C atoms in Mo–Ta alloy was analyzed, and the results showed that Ta atom hinder the diffusion of H atom, but promote the diffusion of C atom. The electronic structure of Mo–Ta alloys was analyzed by the projected density of states (PDOS). On the other hand, the addition of C/H atoms will reduce the strength and improve the ductility of Mo–Ta alloys. For the interstitial atoms, the effect of C on the alloy is stronger than that of H due to the strong electronic interaction between C and Mo–Ta alloy. These results will help us understand the basic behavior of Mo–Ta alloy and the research of the first wall material of fusion reactor.

Effect of Alloying Elements on the Compressive Mechanical Properties of Biomedical Titanium Alloys: A Systematic Review.

Due to problems such as the stress-shielding effect, strength–ductility trade-off dilemma, and use of rare-earth, expensive elements with high melting points in Ti alloys, the need for the design of new Ti alloys for biomedical applications has emerged. This article reports the effect of various alloying elements on the compressive mechanical performance of Ti alloys for biomedical applications for the first time as a systematic review following the PRISMA (Preferred Reporting Items for Systematic Reviews and Meta-Analyses) guidelines on this subject. The search strategy in this systematic review used Scopus, Web of Science, and PubMed databases and searched the articles using (Beta-type OR β) AND Titanium AND (Mechanical property OR Microstructure) AND Alloying element keywords. Original articles from 2016 to 2022 published in English have been selected for this study as per the inclusion criteria. The results have shown that Nb can be used as the primary alloying element with Ti as it is a strong β-stabilizer element which also reduces the elastic modulus of Ti alloys. The β-eutectic elements (Fe, Cr, and Mn) have also emerged as cost-effective alloying elements that could improve the mechanical performance of Ti alloys. Ti–Nb–Zr–Ta alloyed with Si has shown potential to withstand the strength–ductility trade-off dilemma. The combination of a Ti–Nb binary alloy has emerged as an attractive material for designing low elastic modulus Ti alloys. The mechanical performance of the Ti–Nb alloy can be further improved using the β-eutectic (Fe, Cr, and Mn) and neutral (Zr, Sn) elements to be alloyed with a Ti–Nb binary alloy. The strength–ductility trade-off issue can be overcome using Si as an alloying element in Ti–Nb–Zr–Ta alloys.

An antibacterial mechanism of titanium alloy based on micro-area potential difference induced reactive oxygen species

• Micro-area potential difference (MAPD) existed on Ti-M alloy due to the second phase formation. • MAPD played a key factor in the antibacterial properties of titanium alloys. • MAPD enhances the galvanic reaction and electron transfer of Ti-M alloy. • Electron transfer induces ROS in bacteria but does not affect MC3T3 proliferation and adhesion. Antimicrobial material is highly desired because of the increasing demand in biomedical application to prevent from the formation of biofilm. A common strategy for enhancing the antibacterial property of a metal material is to incorporate toxic metal such as Cu and Ag. However, the reported Cu 2+ or Ag + released concentration from antibacterial alloys was much less than the reported minimum inhibitory ion concentrations (MIC), revealing the existence of an unknown alternative antimicrobial mechanism not relying on the toxicity of the metal ions. Herein, we proposed a new antibacterial mechanism that the antibacterial effectiveness of the different alloys is proportional to the micro-area potential differences (MAPDs) on the surface of the alloys. We designed three kinds of Ti-M (M=Zr, Ta and Au) alloys to eliminate the potential antibacterial contribution from Cu and Ag ion. We demonstrated that high MAPDs are associated with great production of reactive oxygen species (ROS), resulting in the killing effect to the biofilm known to be associated with implant infections ( Staphlococcus aureus and Escherichia coli ). These results provide new insights for the design of antibacterial alloys.

Microstructure characterization and evolution mechanism of titanium during laser surface nitriding

Laser surface nitriding technique has been successfully applied to greatly improve the nitriding efficiency of titanium alloy with marginally changing the surface roughness of substrate. However, due to the ultra-short action time, it is hard to understand the evolution mechanism of the microstructure during nitriding process, which is exactly the foundation to better control the quality of the nitride layer. In order to avoid the interference of alloying elements in titanium alloy on diffusivity of nitrogen, a commercial pure titanium TA2 alloy was used as the substrate in this research. Combined with finite element modeling (FEM) technology and microstructure characterizations, evolution mechanism of the microstructures during laser nitriding were systematically analyzed. Results indicate that microstructures of TiN -, α-Ti(N) - and (α’ + α) are formed successively from the surface to the center. In the laser heating stage, TiN phase and stabilized α-Ti(N) phase formed, while in the cooling stage, α’ martensite formed prior to the surrounding α phase, result in continuous orientation transition in α grains. Due to the gradual decrease of nitrogen contents from the surface to the center, α’ martensite laths coarsened with the directional growth characterizations. • Nitriding microstructures with 27 μm depth are formed on TA2 surface by rapid laser nitriding. • Nitriding microstructures consist of the compound layer and the beneath lath zone. • Microstructures of TiN - α-Ti(N) - (α’ + α) are formed from the substrate surface into the alloy. • α’ martensite laths gradually coarsened with the directional growth characterizations.