Configuration Transition Research Articles

Adsorption configuration and assembly structure of vanadyl phthalocyanine molecule on copper oxide layer

In recent years, regulating organic functional molecule has gradually received much attention in the field of materials due to its significant contribution in improving the charge carrier mobility of nanometer optoelectronic device. Molecular configuration and assembly structure of vanadyl phthalocyanine (VOPc) are systemically investigated on pristine and oxidized Cu(110) surface by using low temperature scanning tunneling microscopy. In the initial deposition stage, two molecular adsorption configurations, referring to O-up and O-down, are randomly distributed on the pristine Cu(110) surface. By oxidizing Cu(110) at different oxygen atmospheres and substrate temperatures, two different copper oxide structures are obtained, i.e. CuO-(2×1) and Cu<sub>5</sub>O<sub>6</sub>-c(6×2). The VOPc molecules are then deposited on both surfaces via thermal evaporation. For the CuO-(2×1) surface, contrastly, extended molecular chains form in the initial adsorption and subsequently the VOPc molecules assemble into an ordered molecular film involving both configurations. The VOPc molecules shows two packing orientations with a rotation angle of about 36° relative to each other. On Cu<sub>5</sub>O<sub>6</sub>-c(6×2), the O-down and O-up molecules are isolatedly adsorbed at the initial coverage. As the coverage increases, molecular assembly film gradually forms a parallelogram-shaped unit cell that involves only the O-up molecules. The molecular film exhibits two distinct molecular orientations with a rotation angle of about 42° relative to each other. The dipole-dipole interaction drives the configuration transition from the O-up configuration to O-down configuration. The O-down VOPc molecules of the second layer tend to be adsorbed on the molecular membrane supported by the Cu<sub>5</sub>O<sub>6</sub>-c(6×2) surface. The dipole-dipole interaction between neighboring molecular layers may be responsible for the preferable adsorption of the second-layered molecules. This study suggests the importance of surface oxidization in modifying configurations and orbital distributions of adsorbed molecules that can affect the charge transport in molecular films during fabricating electronic devices.

Achieving ultrahigh uranium/vanadium selectivity of poly(amidoxime) via coupling MXene-enabled strong intermolecular interaction and separated photothermal interface.

Poly(amidoxime) (PAO) has been recognized as the most potential candidate for extracting uranium from seawater, owing to its merits of outstanding uranium affinity, low cost, and large-scale production. Despite remarkable achievements, existing PAO sorbents suffer from unsatisfactory uranium extraction efficiency and selectivity, as imposed by the inherently sluggish uranium adsorption kinetics and inevitable spatial configuration transition of amidoxime, which diminishes uranium affinity. Herein, we discover a facile and integrated design to elaborate a PAO/MXene nanocomposite that delivers ultrahigh and durable uranium/vanadium (U/V) selectivity. The key to our design lies in harnessing MXene-enabled strong intermolecular interactions to PAO to minimize the spatial configuration transition of amidoxime and stabilizing its superior uranium affinity, as well as creating a separated photothermal interface to maximize temperature-strengthened affinity for uranium over vanadium. Such a synergetic effect allows the nanocomposite to acquire over a 4-fold improvement in U/V selectivity compared to that of pure PAO as well as an unprecedented distribution coefficient of uranium compared to most state-of-the-art sorbents. We further demonstrate that our nanocomposite exhibits durable U/V selectivity with negligible attenuation and good antibacterial ability even in long-term operation. The design concept and extraordinary performance in this study bring PAO-based sorbents a step closer to practical uranium extraction from seawater.

Pressure-Modulated Interface Engineering toward Realizing Core@Shell Configuration Transition.

The strained interface of core@shell nanocrystals (NCs) can effectively modulate the energy level alignment, thereby significantly affecting the optical properties. Herein, the unique photoluminescence (PL) response of doped Mn ions is introduced as a robust probe to detect the targeted pressure-strain relation of CdS@ZnS NCs. Results show that the core experiences actually less pressure than the applied external pressure, attributed to the pressure-induced optimized interface that reduces the compressive strain on core. The pressure difference between core and shell increases the conduction band and valence band offsets and further achieves the core@shell configuration transition from quasi type II to type I. Accordingly, the PL intensity of CdS@ZnS NCs slightly increases, along with a faster blue-shift rate of PL peak under low pressure. This study elucidates the interplay between external physical pressure and interfacial chemical stress for core@shell NCs, leading to precise construction of interface engineering for practical applications.

Angle-dependent protrusion of cell membranes

Protrusion induced by cylindrical tubes against cell membranes plays essential roles in numerous biological processes, including filopodia growth, cellular packing or entry of one-dimensional nanomaterials, and indentation of cells by needle-like probes. Though the mechanical interaction between the cell membrane and a perpendicular tube has been widely investigated, little is known about how an inclined protruding tube interacts with the cell membrane. Here, we theoretically investigate the angle-dependent protrusion of cylindrical tubes against cell membranes. It is found that perpendicular protrusion is stabilized by the elastic deformation of cell membranes. Increasing the angle between the protrusion direction and perpendicular direction or increasing membrane tension leads to an increasing peak force for the membrane tubulation and increasing plateau resistive force for the maintenance of membrane tubules. Moreover, two fundamental protruding modes leading to the tip-roof and finial-roof system configurations are identified. Inclined protrusion retards the configurational transition from the tip-roof to the finial-roof configuration, and causes possible bending and buckling of protruding tubes due to a large membrane resistive force. Our results offer fundamental insights into the interaction between cell membranes and one-dimensional nanomaterials, and contribute to the understanding and control of membrane protrusion in biological systems.

The Neogondolella constricta (Mosher and Clark, 1965) group in the Middle Triassic of North America: speciation and distribution

AbstractNeogondolella constricta (Mosher and Clark, 1965) from the Prida Formation at Fossil Hill in central Nevada was the first conodont described from Middle Triassic strata in North America. The species has since been widely reported from elsewhere despite uncertainties about its taxonomic scope and that of similar related taxa. Poor definition of these taxa has spawned a diverse nomenclature and inhibited use of the group in biozonation. Starting with a growth series of topotype N. constricta, we reassess allied contemporaneous taxa from North America. In Nevada, 11 conodont taxa are identified: N. constricta, N. aldae Kozur, Krainer, and Mostler, 1994b, N. cornuta Budurov and Stefanov, 1972, N. ex gr. mesotriassica (Kozur and Mostler, 1982), N. postcornuta (Kovács, 1994), N. posterolonga Kozur, Krainer, and Mostler, 1994b, N. quasiconstricta n. sp., N. quasicornuta n. sp., and three subspecies of N. excentrica Budurov and Stefanov, 1972. Successive associations of taxa display symmetry transition in posterior platform configuration. Subdivision of the upper Anisian–lower Ladinian is provided by dominant N. constricta plus relatively uncommon N. quasiconstricta n. sp. and N. excentrica primitiva n. subsp. in the Rotelliformis ammonoid zone. This is followed in the Meeki through the Subasperum zones by dominant N. cornuta, associated N. posterolonga, plus relatively uncommon N. quasicornuta n. sp., and a variety of asymmetric elements: the Meeki Zone includes N. aff. N. cornuta; the Occidentalis Zone adds N. e. excentrica; and finally, N. e. sigmoidalis n. subsp. appears in the Subasperum Zone. In British Columbia, the three subspecies of N. excentrica are recognized in, respectively, the Deleeni, Chischa, and Matutinum (and younger) ammonoid zones.UUID: http://zoobank.org/7d58c8c8-cd68-498f-bb84-5d43de47f779

Cellular Automata Inspired Multistable Origami Metamaterials for Mechanical Learning.

Recent advances in multistable metamaterials reveal a link between structural configuration transition and Boolean logic, heralding a new generation of computationally capable intelligent materials. To enable higher-level computation, existing computational frameworks require the integration of large-scale networked logic gates, which places demanding requirements on the fabrication of materials counterparts and the propagation of signals. Inspired by cellular automata, a novel computational framework based on multistable origami metamaterials by incorporating reservoir computing is proposed, which can accomplish high-level computation tasks without the need to construct a logic gate network. This approach thus eliminates the demanding requirements for the fabrication of materials and signal propagation when constructing large-scale networks for high-level computation in conventional mechanical logic. Using the multistable stacked Miura-origami metamaterial as a validation platform, digit recognition is experimentally implemented by a single actuator. Moreover, complex tasks, such as handwriting recognition and 5-bit memory tasks, are also shown to be feasible with the new computation framework. The research represents a significant advancement in developing a new generation of intelligent materials with advanced computational capabilities. With continued research and development, these materials can have a transformative impact on a wide range of fields, from computational science to material mechano-intelligence technology and beyond.

Edge radiated temperature fluctuations across confinement regime transitions in favorable and unfavorable drift configurations at ASDEX Upgrade

The physics behind the difference in heating power threshold for the low to high confinement (L–H) transition between favorable ion B×∇B drift (toward active X-point) and unfavorable ion B×∇B drift (away from active X-point) magnetic configurations is a longstanding open question. In this work we show differences in edge ( ρpol=0.95 –1.0) turbulence in matched heating power favorable and unfavorable magnetic configuration discharges using a correlation electron cyclotron emission diagnostic to measure radiated temperature ( Trad ) fluctuations. During power matched L-mode phases, the edge Trad fluctuations have higher amplitude in the unfavorable configuration plasma as compared to the favorable configuration plasma. The lower fluctuation amplitude in favorable configuration is observed concomitant with increased E × B shear, as compared to the unfavorable case. Although this turbulence is different in amplitude, it is similar in radial and frequency structure between favorable and unfavorable configurations and grows to the weakly coherent mode during the I-mode phase of the unfavorable configuration discharge. A broadband pedestal fluctuation is seen during an inter-ELM H-mode period of the favorable configuration discharge.

Defect-dependent mechanical and electrical properties of laser-processed CuO nanowires

Narrow bandgap p-type semiconducting metal oxide nanowires (NWs), such as copper oxide (CuO), have gained significant attention for their potential in the development of electrical nano-devices. Tailoring the mechanical and electrical properties of CuO NWs is crucial for optimizing their functionality in specific applications. In this study, we employ nanosecond laser irradiation to precisely modify the properties of individual CuO NWs by inducing point and line defects, including oxygen vacancies and dislocations. Through controlled laser irradiation, we observe a gradual enhancement in the concentration of oxygen vacancies within CuO NWs until reaching a saturation point. The accumulation of vacancies leads to a substantial residual stress, resulting in lattice distortion and misfit. This high residual stress serves as a catalyst for the nucleation of dislocations, subsequently leading to a meaningful enhancement in plasticity. Remarkably, the density of dislocations demonstrates a strong correlation with the duration of laser irradiation. Prolonged irradiation leads to a thermally activated restoration process, where the dislocation configuration transitions from a random distribution to ordered dislocation loops. Mechanical characterization tests indicate that pristine CuO NWs exhibit brittleness, while laser irradiation renders them ductile with improved plasticity. Furthermore, the laser processing of CuO NWs demonstrates an enhancement in their electrical conductivity and optical absorbance.

Operando structural analysis of phase transition of graphite electrode during Li de-intercalation process using neutron and synchrotron radiation X-ray diffraction

Graphite is widely used as a negative electrode material for lithium-ion batteries. Although it is well known lithium ions are inserted between the graphene layers, the details of the structural transition they undergo are not well understood. In this study, we performed operando structural analyses of graphite electrode during the discharge process using neutron diffraction and synchrotron radiation X-ray diffraction and established a solid foundation to clarify the relationship among the LiCx composition, discharging profile, and phase evolution of lithium-intercalated graphite (LIG). The operando analyses enabled us to assess simultaneously the structural changes of LIG along the ab plane (in-plane structure) and the c-axis direction (stage structure). In the de-lithiation process, the intralayer transition was observed on the ab plane from the LiC6-type to the LiC9-type arrangement near the LiC18 composition. The LiC9-type arrangement was retained in the Li-poorer phases such as LiC27, LiC36, LiC54, and LiC72, where the Li de-lithiation proceeded with the interlayer transition to higher-order stages. The phase transition of LIG involves a rapid switchover between the LiC6-type and the LiC9-type intralayer arrangement along with the changes in the interlayer stage structure. A reaction mechanism for the configurational transition on the intralayer and interlayer structure was updated.

Magnetic-field-driven director configuration transitions in radial nematic liquid crystal droplets.

We study the director configurations of nematic liquid crystal (NLC) droplets with homeotropic anchoring in a magnetic field and report observation of a magnetic-field-driven transition from a deformed radial to an axial-with-defect configuration. Magnetic-field-induced transitions in NLC droplets differ fundamentally from the traditional planar Freedericksz transition due to the spherical droplet geometry and resulting topological defect. This transition has been studied theoretically, but the director configurations and mechanism of defect evolution in an applied magnetic field have yet to be observed experimentally. To this end, we combine polarized optical microscopy with a variable electromagnet (≤1T) for continuous observation of droplet director fields, and we employ Landau-de Gennes numerical simulations to elucidate the director configurations and first-order nature of the transition. We report a configuration transition from point defect to ring defect at a critical field, which varies inversely with droplet radius and is relatively independent of surfactant type and concentration. We also estimate anchoring strengths of commonly used surfactants at the NLC-aqueous interface.

Mechanical properties and thermal stability of ZrCuAlx thin film metallic glasses: Experiments and first-principle calculations

In this work, we provide a holistic picture about the relationship between atomic structure, mechanical properties, and thermal stability of ZrCuAlx thin film metallic glasses (TFMGs) varying the Al content from 0 to 12 at.%, carrying out a broad characterization involving experiments and ab initio molecular dynamic simulations (AIMD). We show that the addition of Al resulted in a change of average interatomic distances by ∼10 pm with the formation of shorter bonds (Al-Zr and Al-Cu), influencing the mechanical response (shear/elastic moduli and hardness) which increases by ∼15% for 12 at.% Al. Moreover, tensile tests on polymer substrate revealed a maximum value for the crack initiation strain of 2.1% for ZrCuAl9, while the strain-to-failure rapidly decreases at higher Al contents. The observed reduction in damage tolerance is correlated to a transition in atomic configuration. Specifically, a maximum in density of full and defective icosahedral cluster population is observed at 9 at.% Al, inducing a more shear-resistant behavior to the material. Thermal stability is investigated by high-energy and conventional x-ray diffraction and electrical resistivity measurements as a function of the temperature. Glass transition (Tg) and crystallization (Tx) temperature increase by Al addition reaching 450 and 500 °C, respectively for ZrCuAl12. The increase in thermal stability is related to the reduction in atomic mobility due to the formation of shorter chemical bonds, inhibiting atomic reconfiguration during crystallization. In conclusion, we provide guidelines to the design of compositional-tailored ZrCuAlx TFMGs with tuned mechanical properties and thermal stability with potential impact on industrial applications.

Final ion yields upon the cascade decay of single K, L, M, and N vacancies in atomic silver

Final ion yields and mean final ion charges upon the cascade decay of single vacancies in the K, L, M, and N shells of atomic silver are calculated using the method of construction and analysis of the decay trees with accounting for the shake-offs caused by the change of atomic core potential due to cascade transitions. The shake-off effect on the final ion yields is found to be comparatively small, the largest increase of the mean final ion charge being 0.20e in the case of initial 2s vacancy. Total semi-relativistic Pauli–Fock energies of cascade-produced ionic configurations were used to calculate cascade transition energies, which made it possible to accurately account for the closing of some non-radiative transitions in multivacancy cascade configurations. For the transitions with overlapping multiplets of initial and final ionic configurations, only energetically allowed transitions between the multiplet components were counted when calculating mean transition energies and transitions branching ratios. Comparison of calculated charge spectra with those reported earlier by other authors demonstrated crucial importance of accurate calculation of transition energies and accounting for multiplet splitting in simulation of cascades in heavy atoms.

Temperature-dependent stacking fault energy, deformation behavior, and tensile properties of a new high-entropy alloy

The stacking fault energy (SFE), deformation behavior, and tensile properties of a new high-entropy alloy (HEA), Fe35Mn35Co10Cr10Ni10 (in at. %), were investigated at room temperature (RT) and −100 °C. Deformation substructure evolution during tensile loading was studied using electron backscattered diffraction (EBSD) and transmission electron microscopy (TEM). A transition in the slip configuration occurred from fully wavy to completely planar with the decrease in temperature from room temperature (RT) to −100 °C. The studied alloy revealed the onset of deformation twinning only during deformation at −100 °C and the same was absent during RT deformation. SFE of the studied alloy at −100 °C was ∼34.2 (±4) mJ/m2. SFE determination required the value of the distance between Shockley partial dislocations and the same was determined employing the transmission electron microscopy (TEM) based weak-beam dark field (WBDF) technique. SFE of the studied alloy could not be estimated at RT due to the presence of closely spaced partial dislocations (perhaps due to very high SFE) and therefore partial separation by the presently employed WBDF technique couldn’t be achieved. The role of friction stress was limited and SFE was revealed as the main factor in defining the active slip mode in HEAs containing no apparent short-range ordering and similar shear moduli. Activation of planar slip-induced dislocation features such as Taylor lattices, microbands, and twinning at −100 °C provided greater work hardening resulting in improved tensile properties compared to RT.

Field‐Induced Multiscale Polarization Configuration Transitions of Mesentropic Lead‐Free Piezoceramics Achieving Giant Energy Harvesting Performance

AbstractThe development of high‐performance (K,Na)NbO3 (KNN)‐based lead‐free piezoceramics for next‐generation electronic devices is crucial for achieving environmentally sustainable society. However, despite recent improvements in piezoelectric coefficients, correlating their properties to underlying multiscale structures remains a key issue for high‐performance KNN‐based ceramics with complex phase boundaries. Here, this study proposes a medium‐entropy strategy to design “local polymorphic distortion” in conjunction with the construction of uniformly oversize grains in the newly developed KNN solid‐solution, resulting in a novel large‐size hierarchical domain architecture (≈0.7 µm wide). Such a structure not only facilitates polarization rotation but also ensures a large residual polarization, which significantly improves the piezoelectricity (≈3.2 times) and obtains a giant energy harvesting performance (Wout = 2.44 mW, PD = 35.32 µW mm−3, outperforming most lead‐free piezoceramics). This study confirms the coexistence of multiphase through the atomic‐resolution polarization features and analyzes the domain/phase transition mechanisms using in situ electric field structural characterizations, revealing that the electric field induces highly effective multiscale polarization configuration transitions based on T–O–R sequential phase transitions. This study demonstrates a new strategy for designing high‐performance piezoceramics and facilitates the development of lead‐free piezoceramic materials in energy harvesting applications.

Spin Glass Behavior in Amorphous Cr2 Ge2 Te6 Phase-Change Alloy.

The layered crystal structure of Cr2 Ge2 Te6 shows ferromagnetic ordering at the two-dimensional limit, which holds promise for spintronic applications. However, external voltage pulses can trigger amorphization of the material in nanoscale electronic devices, and it is unclear whether the loss of structural ordering leads to a change in magnetic properties. Here, it is demonstrated that Cr2 Ge2 Te6 preserves the spin-polarized nature in the amorphous phase, but undergoes a magnetic transition to a spin glass state below 20K. Quantum-mechanical computations reveal the microscopic origin of this transition in spin configuration: it is due to strong distortions of the CrTeCr bonds, connecting chromium-centered octahedra, and to the overall increase in disorder upon amorphization. The tunable magnetic properties of Cr2 Ge2 Te6 can be exploited for multifunctional, magnetic phase-change devices that switch between crystalline and amorphous states.

Nanochannel-Induced Efficient Water Splitting at the Superhydrophobic Interface.

Constructing a favorable reaction configuration at the water/catalyst interface is crucial for high-efficiency semiconductor-based water splitting. For a long time, a hydrophilic surface of semiconductor catalysts has been considered necessary for efficient mass transfer and adequate contact with water. In this work, by constructing a superhydrophobic PDMS-Ti3+/TiO2 interface (denoted P-TTO) with nanochannels arranged by nonpolar silane chains, we observe overall water splitting efficiencies improved by an order of magnitude under both the white light and simulated AM1.5G solar irradiation compared to the hydrophilic Ti3+/TiO2 interface. The electrochemical overall water splitting potential on the P-TTO electrode also decreased from 1.62 to 1.27 V, which is close to the thermodynamic limit of 1.23 V. Through the in situ diffuse reflection infrared Fourier transform spectroscopy, a nanochannel-induced water configuration transition is directly detected. The density functional theory calculation further verifies the lower reaction energy of water decomposition at the water/PDMS-TiO2 interface. Our work achieves efficient overall water splitting through nanochannel-induced water configurations without changing the bulk of semiconductor catalyst, which reveals the significant role of water status at the interface in the efficiency of the water splitting reaction over the properties of catalyst materials.

Tuning of Multi-stability Profile and Transition Sequence of Stacked Miura-Origami Metamaterials

Multi-stable origami structures and metamaterials possess unique advantages and could exhibit multiple stable three-dimensional configurations, which have attracted widespread research interest and held promise for applications in many fields. Although a great deal of attention has been paid to the design and application of multi-stable origami structures, less knowledge is available about the transition sequence among different stable configurations, especially in terms of the fundamental mechanism and the tuning method. To fill this gap, with the multi-stable dual-cell stacked Miura-ori chain as a platform, this paper explores the rules that govern the configuration transition and proposes effective methods for tuning the transition sequence. Specifically, by correlating the energy evolution, the transition paths, and the associated force–displacement profiles, we find that the critical extension/compression forces of the component cells play a critical role in governing the transition sequence. Accordingly, we summarize the rules for predicting the transition sequence: the component cell that first reaches the critical force during quasi-static extension or compression will be the first to undergo a configuration switch. Based on these findings, two methods, i.e., a design method based on crease-stiffness assignment and an online method based on internal pressure regulation, are proposed to tune the stability profile and the transition sequence of the multi-stable origami structure. The crease-stiffness design approach, although effective, cannot be employed for online tuning once the prototype has been fabricated. The pressure-based approach, on the other hand, has been shown experimentally to be effective in adjusting the constitutive force–displacement profiles of the component cells and, in turn, tuning the transition sequence according to the summarized rules. The results of this study will advance the state of the art of origami mechanics and promote the engineering applications of multi-stable origami metamaterials.

Circumventing CO2 Reduction Scaling Relations Over the Heteronuclear Diatomic Catalytic Pair.

In the electrochemical CO2 reduction reaction (CO2RR), CO2 activation is always the first step, followed by the subsequent hydrogenation. The catalytic performance of CO2RR is intrinsically restricted by the competition between molecular CO2 activation and CO2 reduction product release. Here, we design a heteronuclear Fe1-Mo1 dual-metal catalytic pair on ordered porous carbon that features a high catalytic performance for driving electrochemical CO2 reduction to CO. Combining real-time near-ambient pressure X-ray photoelectron spectroscopy, operando 57Fe Mössbauer spectroscopy, and in situ attenuated total reflectance surface-enhanced infrared absorption spectroscopy measurements with density functional theory calculations, chemical adsorption of CO2 is observed on the Fe1-Mo1 catalytic pair through a bridge configuration, which prompts the bending of the CO2 molecule for CO2 activation and then facilitates the subsequent hydrogeneration reaction. More importantly, the dynamic adsorption configuration transition from the bridge configuration of CO2 on Fe1-Mo1 to the linear configuration of CO on the Fe1 center results in breaking the scaling relationship in CO2RR, simultaneously promoting the CO2 activation and the CO release.

Synergistic Role of Temperature and Salinity in Aggregation of Nonionic Surfactant-Coated Silica Nanoparticles.

The adsorption of nonionic surfactants onto hydrophilic nanoparticles (NPs) is anticipated to increase their stability in aqueous medium. While nonionic surfactants show salinity- and temperature-dependent bulk phase behavior in water, the effects of these two solvent parameters on surfactant adsorption and self-assembly onto NPs are poorly understood. In this study, we combine adsorption isotherms, dispersion transmittance, and small-angle neutron scattering (SANS) to investigate the effects of salinity and temperature on the adsorption of pentaethylene glycol monododecyl ether (C12E5) surfactant on silica NPs. We find an increase in the amount of surfactant adsorbed onto the NPs with increasing temperature and salinity. Based on SANS measurements and corresponding analysis using computational reverse-engineering analysis of scattering experiments (CREASE), we show that the increase in salinity and temperature results in the aggregation of silica NPs. We further demonstrate the non-monotonic changes in viscosity for the C12E5-silica NP mixture with increasing temperature and salinity and correlate the observations to the aggregated state of NPs. The study provides a fundamental understanding of the configuration and phase transition of the surfactant-coated NPs and presents a strategy to manipulate the viscosity of such dispersion using temperature as a stimulus.

Duality cascades and parallelotopes

Duality cascades are a series of duality transformations in field theories, which can be realized as the Hanany–Witten transitions in brane configurations on a circle. In the setup of the Aharony–Bergman–Jafferis–Maldacena theory and its generalizations, from the physical requirement that duality cascades always end and the final destination depends only on the initial brane configuration, we propose that the fundamental domain of supersymmetric brane configurations in duality cascades can tile the whole parameter space of relative ranks by translations, hence is a parallelotope. We provide our arguments for the proposal.