Polyhedral Oligomeric Silsesquioxane Cages Research Articles

Cage-rearranged and cage-intact syntheses of azido-functionalized larger T10 and T12 POSSs.

Herein, we investigate the product type and distribution during the synthesis of azido-functionalized larger polyhedral oligomeric silsesquioxanes (POSSs) using 3-chloropropyl- and chloromethyldimethylsilylethyl-functionalized T8, T10, and T12 POSSs as precursors. Our findings indicate that cage rearrangement occurs for the 3-chloropropyl-functionalized POSS cages with a stability order of T12 > T10 > T8, while the chloromethyldimethylsilylethyl-functionalized POSS cages remain structurally intact after the nucleophilic substitution.

Construction of silsesquioxane and phosphinum‐based ionic porous hypercrosslinked polymers for efficient heterogeneous catalytic CO2 cycloaddition

AbstractIn this work, we reported a facile one‐pot route to construct polyhedral oligomeric silsesquioxane (POSS) and phosphonium‐based ionic porous hypercrosslinked polymers (denoted as P‐iPHCPs) with multiple catalytic active sites, which could be regarded as efficient heterogeneous catalysts for the conversion of carbon dioxide (CO2) to highly value‐added cyclic carbonates. The targeted polymers P‐iPHCPs were prepared from a rigid building block octavinylsilsesquioxane (VPOSS) and a low‐cost commercially available organic ionic salt methyltriphenylphosphonium bromide ([Ph3PMe]Br) by the AlCl3‐catalyzed Friedel‐Crafts reaction. The series of polymers P‐iPHCPs that synthesized by different molar ratios of VPOSS to [Ph3PMe]Br possess hierarchical micro/mesoporous structures with high surface areas ranging from 376 to 429 m2 g−1 and behave the flower‐like nano‐morphology. Especially, the desired multiple active sites involved the mixed anions including free Cl−, Br− anions and the in‐situ formed metal‐halogen complex anion [AlCl3Br]− within phosphonium segments, and the enriched POSS‐derived SiOH hydrogen bond donor (HBD) groups that derived from the breakage of POSS cages during the synthetic process. The typical polymer P‐iPHCP‐14 has the aforementioned nucleophilic‐electrophilic multiple catalytic active sites, which was regarded as an efficient recyclable heterogeneous catalyst for synergistic catalytic conversion of CO2 with various epoxides into cyclic carbonates under mild conditions.

Molecular Dynamics Simulation of Polyhedral Silsesquioxane Nanoparticles on Cathodes of Polymer Electrolyte Membrane Fuel Cells

The effect of octahedral cage-structured polyhedral oligomeric silsesquioxane (POSS) on the cathode catalyst layer (CCL) for polymer electrolyte membrane fuel cells (PEMFCs) is investigated via molecular dynamics simulations. A model of CCL is constructed consisting of Pt/C substrate/ionomer–POSS/water–gas. The pair correlation function analysis shows that the dense cover of ionomer chains on the Pt surface and the interaction between sulfonate groups and water molecules are weakened by POSS. At high hydration, the effective water content in the ionomer–POSS phase is drastically reduced with increasing POSS content. Interestingly, at low hydration, the effective water content is slightly improved as the POSS content increases. This water retention effect is probably due to the trapping of water molecules in the POSS cage. The diffusion coefficients of O2 and H3O+ are effectively increased. The method of inserting nanocages into ionomer can avoid the formation of a high-density ionomer film at the Pt/ionomer interface. Moreover, POSS-modified CCLs possess a self-balance of water content, thus achieving enhanced multicomponent transport involving proton, water, and oxygen transfer. It provides a strategy for designing high-performance PEMFC, particularly for fuel cell vehicles due to the self-balance property under changing humidity.

Theoretical analysis of substituent- and cage-dependent electronic properties of POSS

Polyhedral Oligomeric Silsesquioxanes (POSS) molecules have emerged as promising alternatives to traditional silica nanoparticles and organosilicon molecules due to their ability to attach a variety of substituents to their central siloxane cages. The electronic features of POSS are dependent on the structures of substituents and cages, making them attractive for applications, such as drug nanocarriers, chemosensors, Janus particles, and inorganic–organic nanocomposites. However, the lack of detailed geometric data poses a difficulty in extending POSS studies using the force field method or beyond the quantum mechanical level. To lay a foundation for future experimental and computational studies, we present new theoretical insights into the influence of substituent and cage on the cage geometries (Si–O bond length, pore width, cage volume, etc.) of 16 substituted POSS molecules with various substituents and cages (T7, T8, T10, T12). Our results show that the mean Si–O bond length of the POSS cage can be in the range of 1.619–1.670 Å depending on the substituent and cage. Moreover, we show that the expansion or shrinkage of POSS depends on the types of substituents, rather than the inductive effect of substituents. Our results demonstrate that the conformational stability, dipole moment, and polarizability of POSS depend on the substituents’ size, substituents’ shape, type of functional moieties of substituents, and cage size. In addition, the HOMO-LUMO bandgap of POSS can be tuned by changing the hydrocarbon chain length, number of aromatic rings, and types of functional groups on the substituents. Finally, we report several sets of geometric data that are transferable to the existing parametrization methods of force field models.

Sequential self-assembly and self-coacervation actuate water-triggered robust bonding: From universal underwater adhesion to on-demand detachable bioadhesion

Achieving strong and long-lasting underwater adhesion remains a great challenge because both interfacial hydration film and continuous water erosion significantly impede the interfacial bonding. Herein, a sequential self-assembly and self-coacervation strategy was proposed to actuate water-triggered robust underwater bonding to both abiotic materials and biological tissues. With the embedded polyhedral oligomeric silsesquioxane (POSS) cages serving as rigid and hydrophobic segments, the amphiphilic hybrid hyperbranched polymer featured conformation variation capacity under the synergy of sharply decreased solubility and cluster effect of polyfunctional groups on POSS cages. The polymer firstly altered conformation into an assembled adhesive with extremely rigid hydrophobic POSS clusters for significantly enhancing the cohesion. Once touching water, the prompt aggregation of POSS clusters and followed self-coacervation behavior of the adhesive boosted interfacial water expulsion and catechol groups exposure, resulting in spontaneous solidification and universally eminent underwater adhesion between the adhesive and diverse substrates. The adhesive not only exhibited outstanding adhesive performances in water, seawater, PBS, acid, and alkali solutions, but also possessed excellent anti-erosion capacity to demonstrate long-lasting underwater adhesion for months. Prominently, a smart bioadhesive was developed via introducing tissue adhesive group and sensitive disulfide bond, which revealed strong adhesion to wet tissues and on-demand debonding capacity. The water-triggered robust and long-lasting underwater bonding mediated by sequential self-assembly and self-coacervation opens up a new horizon in the design and progress of smart and high-performance underwater adhesives.

Linker engineering of larger POSS-based ultra-low-k dielectrics toward outstanding comprehensive properties

Low-dielectric-constant (low-k) materials are an indispensable part of microprocessors as they can alleviate electronic crosstalk, charge build-up, and signal propagation delay. However, existing low-k materials usually have k values higher than 2 and inferior thermo-mechanical properties, which restrict the development of microelectronic devices. Although we have recently discovered that larger polyhedral oligomeric silsesquioxanes (POSS) are useful building blocks for fabricating advanced ultra-low-k materials, it remains to be seen whether a POSS cage and linker could be leveraged to tune the structure-property of the materials and create ultra-low-k materials with improved comprehensive properties. Herein, we propose a series of POSS-based hybrid materials (i.e., c-TnPBn and c-TnFn, n = 8, 10, and 12) that consist of distinct POSS cages and linkers. When the linker is kept identical, the gradual enlargement of the POSS cage enhances the porosity/free volume fraction of materials, resulting in quasi-linearly reduced k values for the resulting materials (k = 1.93 for c-T12PB12, 2.14 for c-T12F12). Meanwhile, the materials’ comprehensive properties can be significantly improved by increasing the POSS cage size or varying the linker’s length and type. As a result, ultra-low-k materials with good processability, low surface roughness (< 0.40 nm), excellent thermostability (> 480 °C), mechanical properties (elastic modulus > 2.5 GPa), and hydrophobicity have been obtained, and the low-k values can be maintained under high temperature (e.g., 300 °C) and wet condition. This work reveals the critical contribution of POSS cage size and linker to the structure and properties of POSS-based low-k materials and offers promising materials for the future of the microelectronic industry.

PEG-linked bis-imidazolium and polyhedral oligomeric based silsesquioxanes as ion exchangers

There is a growing demand for functional silica-based porous materials in the market. In this regard, PEG-linked bis-imidazolium chloride silsesquioxane (PIMAILS, an ionic liquid) was functionalized on polyhedral oligomeric silsesquioxanes (POSS). The materials were characterized with FT-IR, 1H and 13C NMR. The structural morphology of the adsorbent was analyzed with BET, TEM and FE-SEM techniques. The NMR results imply that during the PIMAILS loading over the POSS framework, the O–CH3 bond was cleaved out and the ionic liquid bridge was attached to the Si vertices of the POSS cage. The BET surface area of 853 m2 g−1 indicates higher porosity of the material. The ion exchanger has smooth and round-shaped morphology with continuous porous structures and an average particle size of 20–40 nm. The porous framework contains imidazolium binding sites, which have ion-exchange capability and was used to separate Cr(VI) ion from wastewater. With neat POSS and POSS-IL, comparative adsorption of Cr(VI) was carried out at different contact times, wastewater pH and initial concentrations. Maximum removal of 92% of Cr(VI) from an aqueous solution was achieved with POSS-IL at pH 2. The results showed that the adsorption adequately fit the linear Langmuir model and second-order kinetics. After adsorption, the adsorbents and treated solutions were further analyzed with XPS and ion chromatography instruments, indicating ion exchange mediated adsorption for the POSS-IL.

Solvent-Free Templated Synthesis of Core-Crosslinked Star-Shaped Polymers in Supramolecular Body-Centered Cubic Phase.

This study reports the exploration of a solvent-free supramolecular templated synthesis strategy toward highly core-cross-linked star-shaped polymers (CSPs). To achieve this, a kind of cross-linkable giant surfactant, based on a functionalized polyhedral oligomeric silsesquioxanes (POSS) head tethered with a diblock copolymer tail containing reactive benzocyclobutene groups, is designed and prepared. By varying the volume fraction of linear block copolymer tail, these giant surfactants can self-assemble into a body-centered cubic (BCC) structure in bulk, in which the supramolecular spheres are composed of a core of POSS cages, a middle shell of crosslinkable poly(4-vinylbenzocyclobutene) (PBCB) blocks, and a corona of inert polystyrene (PS) blocks. The solvent-free thermally induced cross-linking reaction of the benzocyclobutene groups can be finished in 5min upon heating, resulting in well-defined polymeric spheres with over 90 linear chains surrounding the cross-linked cores. The outer PS blocks serve as the protection corona to ensure that cross-linking of giant surfactants occurs in each supramolecular spherical domain. Given the modular design and diversity of the POSS-based giant surfactants, it is believed that the strategy may enable access to a wide range of CSPs.

Self-Healing Silsesquioxane-Based Materials

This review is devoted to self-healing materials (SHM) containing polyhedral oligomeric silsesquioxanes (POSS) as building blocks. The synthetic approach can vary depending on the role POSS are expected to play in a given system. POSS (especially double-decker silsesquioxanes) can be grafted in side chains of a polymer backbone or used as segments of the main chain. Appropriate functionalization allows the formation of dynamic bonds with POSS molecules and makes them an active component of SHM, both as crosslinking agents and as factors that enhance the dynamics of macromolecules in the polymer matrix. The latter effect can be achieved by reversible release of bulky POSS cages or by the formation of separated inclusions in the polymer matrix through hydrophobic interactions and POSS aggregation. The unique properties of POSS-based self-healing systems make them interesting and versatile materials for various applications (e.g., repairable coatings, sealants, sensors, soft materials for tissue engineering, drug delivery, and wound healing).

Progress in the self-assembly of organic/inorganic polyhedral oligomeric silsesquioxane (POSS) hybrids.

This Review describes recent progress in the self-assembly of organic/inorganic POSS hybrids derived from mono-, di-, and multi-functionalized POSS cages. We highlight the self-assembled structures and physical properties of giant surfactants and chain-end- and side-chain-type hybrids derived from mono-functionalized POSS cages; main-chain-type hybrids derived from di-functionalized POSS cages; and star-shaped hybrids derived from multi-functionalized POSS cages; with various polymeric attachments, including polystyrene, poly(methyl methacrylate), phenolic, PVPh, and polypeptides.

POSS-graphene oxide nanocomposite membranes for ethanol permselective pervaporation

Pervaporation has shown great promise in recovering alcohol from fermentation broth. However, the separation performance determined by membrane material and structure should be further improved. In this study, polyhedral oligomeric silsesquioxane (POSS) nanocage was used as nanofillers to provide preferential pathways for ethanol molecules in polydimethylsiloxane (PDMS) membrane. To improve dispersion of POSS nanocages in PDMS matrix, the POSS were anchored on the graphene oxide (GO) nanosheets. At the meantime, the aggregation of GO nanosheets could also be prevented as the GO surface turned to be hydrophobic by the modification of POSS, and the nanoporous POSS cage could reduce interface defects between GO nanosheets and PDMS matrix. The obtained POSS-GO/PDMS nanocomposite membranes exhibited excellent separation performance for ethanol permselectivity. The flux could reach 1346.9 g m −2 h −1 with a separation factor of 11.2 when recovery of ethanol from 5 wt% ethanol/water mixture. In addition, the separation performance of the POSS-GO/PDMS membrane could maintain stability in 120 h. Therefore, the POSS modified GO nanosheets shown great potential in the fabrication of ethanol permselective pervaporation membranes. • POSS-GO nanocomposites were fabricated and incorporated into the PDMS matrix for preparation of pervaporation membranes. • The nanoplates can be well dispersed in PDMS as the hydrophobic POSS has been uniformed anchored to the GO nanosheets. • The obtained POSS-GO/PDMS membranes exhibited excellent pervaporation performance for ethanol permselectivity.

Fluorine‐Free Nanoporous Low‐k Dielectric Film Covalently Grafted on Si via Aryldiazonium Chemistry

AbstractFluorine‐free nanoporous low‐k dielectric film containing polyhedral oligomeric silsesquioxanes (POSS) cage nanostructure is covalently grafted onto a p‐Si(100) surface via aryldiazonium chemistry, in an open environment without any inert gas protection. The method is based on two‐step reduction of aryldiazonium salts. First, 4‐nitrobenzene diazonium tetrafluoroborate (NBD) is reduced and a polynitrophenyl (PNP) passivation layer is formed on the p‐Si(100) surface, in an aqueous media with hydrofluoric acid being added. Second, NBD is reduced by electrons provided by external pulse power supply, and initiates octavinylsilsesquioxane polymerizing to form the film, which is composed of PNP and massive POSS cage, leading to a low k value. The average dielectric constant of the film is 2.12 at the range of 600 to 1700 nm wavelength, much lower than some low‐k materials reported by recent literature and traditional materials. The prepared films completely cover the Si substrate and are compact and thickness controllable. In particular, the k value can be controlled though the concentration of NBD. The low‐k properties imply they will be suitable to be utilized as high‐performance dielectric materials in future higher frequency communication field.

Shape recovery and reprocessable polyurethanes crosslinked with double decker silsesquioxane via Diels-Alder reaction

In this work, a novel organic-inorganic polyurethane thermoset crosslinked with a difunctional polyhedral oligomeric silsesquioxanes (POSS) was synthesized via Diels-Alder (DA) reaction. First, a linear multi-segmented polyurethane with furan rings in the main chain (PU–F) was synthesized via the step-growth polymerization of 4,4′-diphenylmethane diisocyanate (MDI) with poly(tetramethylene ether glycol) (PTMEG) and 2,5-furandimethanol. Second, 3,13-dimaleimido octaphenyl double decker silsesquioxane (3,13-dimaleimido DDSQ) was synthesized and used as the crosslinker of the linear PU-F. By adjusting the feed ratio of the difunctional POSS macromer (i.e., 3,13-dimaleimido DDSQ) to PU-F, the organic-inorganic PU thermosets (denoted DAPU-DDSQs) with variable densities of crosslinking were obtained. It was found that the DAPU-DDSQs were microphase-separated and that POSS cages were aggregated into the spherical microdomains, the sizes of which were 20–40 nm in diameter. It was demonstrated that the crosslinking in the DAPU-DDSQs had two aspects of contribution: i) the chemical crosslinking with D-A adducts between 3,13-dimaleimido DDSQ and PU-F and ii) the physical crosslinking with POSS microdomains via the aggregation of POSS cages. The DAPU-DDSQs displayed the improved thermomechanical properties with increasing 3,13-dimaleimido DDSQ incorporation. Owing to the formation of crosslinked structures, the DAPU-DDSQs displayed excellent shape memory properties. More importantly, the DAPU-DDSQs displayed the reprocessing (or recycling) properties at the temperature of retro DA reaction. The reprocessing properties of the DAPU-DDSQs can be utilized to re-edit the original shapes of shape memory PUs.

Thermal ablation mechanism of polyimide reinforced with POSS under atomic oxygen bombardment

The thermal stability and reactive atomic oxygen (AO) resistance of polyhedral oligomeric silsesquioxane (POSS)-reinforced polymer nanocomposites have attracted significant interest from both experimental and theoretical researchers. However, understanding the damage-mitigation mechanism of inorganic core constituents in POSS hybrids is still limited due to the lack of empirical insights. In this study, the ablation resistance of polyimide (PI)/POSS nanocomposites was examined using reactive molecular dynamics and first-principles calculations in terms of physicochemical changes in the POSS cage structure. According to the AO collision kinetic energy and frequency level, the origin of ablation resistance in the system is classified. Each damage-mitigation behavior is strongly correlated with the normalized residue mass and byproduct distribution during the erosion of the surface. In particular, the results suggest that the kinematic collapse of POSS prevents the exfoliation of large carbonic molecules and plays a crucial role in the formation of a ceramic passivation layer, which significantly improves the damage-mitigation effect. Therefore, this study provides a new perspective on the design of ablative thermal protection systems by understanding the behavior of the inorganic nanocage constituting POSS hybrids at the molecular level.

Modification of Thiol-Ene Ionogels with Octakis(methacryloxypropyl) Silsesquioxane.

The effect of polyhedral oligomeric silsesquioxane (POSS) on the synthesis and properties of hybrid organic–inorganic ionogels was investigated using octakis(methacryloxypropyl) silsesquioxane (methacryl-POSS). Ionogels were prepared in situ by thiol-ene photopolymerization of triallyl isocyanurate with pentaerythritol tetrakis(3-mercaptopropionate) in a mixture of imidazolium ionic liquid 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide (EMImNTf2) and propylene carbonate (PC). Investigations included the kinetics of hybrid materials formation and selected physical and mechanical properties. The disadvantage of ionogels without the methacryl-POSS modifier is leakage and insufficient mechanical properties. Modifying the thiol-ene matrix by the addition of methacryl-POSS made it possible to obtain non-leaking ionogels with improved mechanical and conductive properties. The steric hindrance of POSS cages and high-density network formation played important roles in ionogel synthesis: decrease of polymerization rate (with almost no effect on conversion), as well as dimensions of the formed polymer spheres during dispersion polymerization (highly cross-linked polymer has poorer solubility in polymerizing medium at a similar conversion, and nucleation begins at lower conversion), an increase of glass transition temperature and puncture strength. Hybrid ionogels with high ionic conductivity in the range of 4.0–5.1 mS∙cm−1 with the maximum parameter for 1.5 wt.% addition of the methacryl-POSS were obtained, which can be associated with ion-pair dissociations in ionic liquid clusters caused by methacryl-POSS.

A POSS-assisted fluorescent probe for the rapid detection of HClO in mitochondria with a large emission wavelength in dual channels.

Hypochlorous acid (HClO) is closely related to many diseases and is an inevitable part of the physiological processes. It is significant to detect HClO in mitochondria for getting meaningful physiological and pathological information. However, adequate tools to detect HClO with emissions in two channels are rarely reported. To achieve this target, in this work, a "turn-off" visual and near infrared (NIR) fluorescent dual emission probe D6 based on polyhedral oligomeric silsesquioxanes (POSS) was successfully designed and synthesized. D6 showed high selectivity and sensitivity to HClO. Notably, the emission wavelength of D6 reached 820 nm due to the assistance of the POSS cage. In addition, bioimaging experiments clearly showed that probe D6 promoted the visualization of exogenous and endogenous HClO in living HepG2 cells and zebrafish models.

Improving superhydrophobicity of polydimethylsiloxanes using embedding fluorinated polyhedral oligomeric silsesquioxanes cages

Improving the superhydrophobicity of polydimethylsiloxane (PDMS) is a current interest in a wide range of applications from biomedical to aerospace. Although many fabrication techniques are available to improve the superhydrophobicity of PDMS, a significant problem occurs when the fabrication technique applies as a scalable but simple one. Here, we have described simple methods to achieve superhydrophobicity of PDMS using short-chained fluorinated polyhedral oligomeric silsesquioxanes (FPOSS). Two species of FPOSS were incorporated into PDMS using four different methods; non-solvent blending, solvent blending, spraying FPOSS/PDMS solution onto a partially cured PDMS matrix and spraying only FPOSS solution onto partially cured PDMS surfaces. Among two FPOSS species, spraying FPOSS onto partially cured PDMS produced a superhydrophobic surface with a static water contact angle of 167° ± 1°. Trifluoropropylisobutyl POSS (TFP) resulted in a higher hydrophobicity than trifluoropropyl POSS cages (CM). The multi-scale structured surface morphology, compatibility of functional groups attached to FPOSS and the fluorine content have shown a significant contribution on the superhydrophobicity in FPOSS/PDMS systems. Amorphous nature of the PDMS has improved upon incorporating FPOSS. Hence, this work presents a detailed study on the effect of the preparation method of FPOSS/PDMS composite on its superhydrophobicity.

Parallel-Mode EPR of Atomic Hydrogen Encapsulated in POSS Cages

In a typical EPR experiment, the transitions require that the static magnetic field $$B_0$$ is oriented perpendicular to the microwave field $$B_1$$ (perpendicular mode). This is determined by the transition rules either in the classical or in the quantum mechanical description. However, there are cases where EPR transitions are observed when $$B_0$$ is oriented parallel to $$B_1$$ (parallel mode). Quite numerous studies can be found in the literature where EPR transitions in both modes (dual-mode EPR) are feasible. In the majority of cases, dual-mode EPR studies are typically applied in $$S>1/2$$ systems where non-zero transition probabilities for the parallel mode are the result of the state mixing provided by the zero-field splitting interaction. On the other hand, the observation of parallel-mode EPR signals in $$S=1/2$$ systems becomes feasible when strong hyperfine interaction between the electronic and nuclear spin is present, as has been theoretically predicted for the hydrogen atom having a hyperfine coupling constant of $$A_0=1420$$ MHz (Weil in Concepts Magn Reson Part A 28:331, 2006). Herein, we report the first dual-mode X-band EPR experiments of hydrogen atom (both isotopes $$^1$$ H and $$^2$$ H) encapsulated in polyhedral oligomeric silsesquioxane cages. We extend the theory to the case of deuterium and we extract analytical formulas for transition probabilities. For the forbidden transitions, this study revealed a first-order dependence of resonance fields on the nuclear g-factor, $$g_{\mathrm{n}}$$ , and the existence of a clock transition with $$f=307$$ MHz.

Molecular Patchy Clusters with Controllable Symmetry Breaking for Structural Engineering.

Anisotropic patchy particles with molecular precision are exquisite building blocks for constructing diverse meso-structures of high complexity. In this research, a library of molecular patchy clusters consisting of a collection of functional polyhedral oligomeric silsesquioxane cages with exact regio-configuration and composition were prepared through a robust and modular approach. By meticulously tuning the composition, molecular symmetry, and other parameters, these patchy clusters could assemble into diverse nanostructures, including unconventional complex spherical phases (i.e., Frank-Kasper σ phase and dodecagonal quasicrystalline phase). As the size of the hydrophilic patch expands, a transition sequence from disorder to hexagonally packed cylinders and then to double gyroids was recorded, corresponding to a progressive decrease of interfacial curvature. On the other hand, regioisomers with the same composition but different regio-configuration adopt similar molecular packing but varied phase stability, as a result of the local self-sorting process to alleviate excess unfavorable interfacial contact. These precisely defined molecular patchy clusters provide a model system for a general understanding of the hierarchical structure formation and evolution based on anisotropic spherical building blocks at the nanoscale.

Constituent Isomerism-Induced Quasicrystal and Frank–Kasper σ Superlattices Based on Nanosized Shape Amphiphiles

Naturally, subtle variations in the chemical structures of constituent molecules may significantly affect their multiscale spatial arrangements, properties, and functions. Deceptively simple spheri...