Formation Of Coordination Polymers Research Articles

Homochiral metal-organic frameworks based on amino acid ligands for HPLC separation of enantiomers.

Natural amino acids are well known to form coordination polymers with transition metal ions. In this study, six homochiral metal-organic frameworks constructed from Zn2+ or Co2+ ions and various enantiopure amino acid (L-tyrosine, L-histidine, L-tryptophan and L-glutamic acid), namely [Zn(L-tyr)]n (L-tyrZn), [Zn4 (btc)2 (Hbtc)(L-His)2 (H2 O)4 ]·1.5H2 O, {[Zn2 (L-trp)2 (bpe)2 (H2 O)2 ]·2H2 O·2NO3 }n , [Co2 (L-Trp)(INT)2 (H2 O)2 (ClO4 )], [Co2 (sdba)((L-Trp)2 ] and [Co(L-Glu)(H2 O)·H2 O]∞ , were synthesized according to the methods previously reported in the literature. The six homochiral MOFs were explored as the chiral stationary phases for high-performance liquid chromatographic separation of enantiomers using hexane/isopropanol or hexane/dichloromethane as mobile phase. Various types of enantiomers such as alcohols, amines, ketones, ethers, organic acids, etc. can be resolved on these homochiral MOF columns. The results revealed that the enantioseletivities of homochiral MOFs based on amino acids as chiral bridging ligands used as stationary phases are practical in HPLC.

1,3-Dithiolane and 1,3-Ferrocenyl-dithiolane as Assembling Ligands for the Construction of Cu(I) Clusters and Coordination Polymers

The parent compound 1,3-dithiolane L1 reacts with CuI providing the 2D coordination polymer (CP) [{Cu(μ2-I)2Cu}(μ2-L1)2] n (CP1). The Cu⋯Cu distance within the Cu(μ2-I)2Cu rhomboid is temperature-dependent and shrinks from 2.9081(7) (294 K) to 2.8743(5) (100 K). The 1D polymeric compound [{Cu(μ2-Br)}(μ2-L1)] n (CP2) was isolated upon treatment of CuBr with L1 in MeCN solution. Likewise, treatment of L1 with CuCl in a 1:1 ratio results in formation of the unusual 1D CP [{Cu(μ2-Cl)}(μ2-L1)] n (CP3). In contrast to CP1 incorporating dinuclear Cu(μ2-I)2Cu units as connection nodes, the isolated Cu atoms of CP2 and CP3 are bridged by a single μ2-Br or μ2-Cl ligand giving rise to infinite [Cu(μ2-X)Cu] n ribbons. Reaction of ferrocenecarbaldehyde with 1,2-ethanedithiol affords the crystallographically characterized ferrocenyl-dithiolane Fc-C(H)S2C2H4 (L2). L2 was used as organometallic dithioether ligand to assemble, upon treatment with CuI, the original luminescent 1D CP [{Cu2(μ4-I)(μ2-I)}(μ2-L2)] n (CP4), in which both S atoms of one L2 molecule span two copper centers of a meander-shaped infinite (CuI) n ribbon. Upon reaction of L2 with CuBr in a 1:2 ratio, the discrete tetranuclear cluster [{Cu4(μ2-Br)2(μ3-Br)2}(μ2-L2)2(MeCN)2] (CL1) is formed.

One-dimensional coordination polymers based on [Ru2(DMBA)4]2+ units

The diruthenium(III) coordination polymers [Ru2(DMBA)4(μ-O,O′-WO4)·2H2O]∞ (1) and [Ru2(DMBA)4(μ-O,O′-1,4-(O2C)2C6H4)·2H2O]∞ (2) were obtained through slow diffusion of Ru2(DMBA)4(NO3)2 in acetonitrile into an aqueous solution containing salts of WO42− and terephthalate, respectively. Both new compounds were characterized using single crystal X-ray diffraction, which revealed the retention of the essential geometrical features of the [Ru2(DMBA)4]2+ units upon the formation of coordination polymers. The temperature-dependence of the magnetic susceptibilities of compounds 1 and 2 and their precursors, Ru2(DMBA)4Cl2 and Ru2(DMBA)4(NO3)2, were measured from 5 to 300K. Fittings of the χ vs. T data indicate that the magnetic behaviors are mostly due to the zero-field splitting of the S=1 Ru2 units and there is only a minimal inter-unit coupling in 1 and 2.

Synthesis, characterization and antioxidant/cytotoxic activity of oxovanadium(IV) complexes of methyliminodiacetic acid and ethylenediaminetetracetic acid

Two oxovanadium(IV) complexes, viz., [VO(Me-IDA)(H2O)2] (1) and NaH[VO(EDTA)]·4H2O (2) (Me-IDA = methyliminodiacetic acid and EDTA = ethylenediaminetetraacetic acid) have been synthesized and characterized by FT-IR, UV–Vis, mass spectrometry, elemental analysis, magnetic moment and thermal analysis, as well as electrochemical measurements including cyclic voltammetry. Both compounds are monomeric with distorted octahedral geometries. Compound 2 has been structurally characterized by using X-ray crystallography. It shows an octahedral V(O)N2O3 coordination geometry, which exhibits chemically significant hydrogen bonding interactions besides showing coordination polymer formation. Compounds 1 and 2 show an irreversible redox peak around +0.80 V versus Ag/AgCl corresponding to one-electron oxidation of V(IV) to V(V). The free radical scavenging activity of compounds 1 and 2 were done using 2,2-diphenyl- 1-picrylhydrazyl (DPPH). Both compounds have shown encouraging ROS scavenging activities. The cytotoxicity effects of both compounds toward two different tumor cells (HePG2 and MCF-7) have been also studied by MTT assay. The IC50 values obtained, after 48 h incubation at 37 °C for HepG2 and MCF-7 cell lines were 74.23 and 42.04 μg/mL for compound 1 and 65.56 and 48.34 μg/mL for compound 2, respectively. Conclusively, the present investigation provides preliminary results which suggest that such compounds can be promising alternative antitumor agents.

Selective Crystallization of Phosphoester Coordination Polymer for the Separation of Neodymium and Dysprosium: A Thermodynamic Approach.

Thermodynamics of the formation of coordination polymers (CPs) or metal-organic frameworks (MOFs) has not been focused on, whereas many CPs or MOFs have been synthesized in a solution. With a view of separating Nd3+ and Dy3+ in an aqueous solution, we demonstrate that crystallization of the CPs of Nd3+ and Dy3+ based on dibutyl phosphoric acid (Hdbp) can be thermodynamically described; crystallization yields of [Ln(dbp)3] (Ln = Nd or Dy) complex are predicted well using a simple calculation, which takes the apparent solubility products (Kspa) for [Ln(dbp)3] and the acid dissociation constant of Hdbp into account. The Kspa values of [Nd(dbp)3] and [Dy(dbp)3] are experimentally determined to be (1.3 ± 0.1) × 10-14 and (2.9 ± 0.4) × 10-18 M4, respectively, at 20 °C. The ratio of these Kspa values, that is, ca. 4500, is significantly larger than the ratio of the solubility products for inorganic salts of Nd3+ and Dy3+. Therefore, Nd3+ and Dy3+ are selectively crystallized in an aqueous solution via the formation of CPs. Under optimized conditions, Dy3+ crystallization is preferable, whereas Nd3+ remains in the solution phase, where the ratio of the Dy molar content to the total metal content (i.e., Nd + Dy) in the crystal is higher than 0.9. The use of acids, such as HCl or HNO3, has no practical impact on the separation in an aqueous solution.

Supramolecular Open-Framework of a Bipyridinium-Carboxylate Based Copper Complex with High and Reversible Water Uptake

The rigid zwitterionic ligand 1,1′-bis(4-carboxylatphenyl)-(4,4′-bipyridinium) (pc1) and copper(II) ions give rise to a linear complex [Cu(pc1)2(H2O)4]2+, which self assembles in a pseudo tetragonal supramolecular arrangement leading to the supramolecular open-framework [Cu(pc1)2(H2O)4](Cl)2·8H2O exhibiting an open structure including water molecules and chlorides in pores. The dehydration of this material occurs at relatively low temperature (70 °C) and results in structure modification accompanied by shrinking of the crystals. Characterization of the obtained material by FT-IR spectroscopy reveals appearance of coordinated carboxylates groups, which may indicate the formation of coordination polymer after dehydration. Water adsorption (maximum uptake 0.23 g H2O per g) on dehydrated material allows to restore the initial structure. A large hysteresis in water adsorption-desorption isotherm is characteristic of a significant modification of the structure during the hydration-dehydration cycle which is in line with the structural transition determined from thermodiffractometry and FT-IR spectroscopy.

Arene Selectivity by a Flexible Coordination Polymer Host.

The coordination polymers [Ag4(O2CCF3)4(phen)3]⋅ phen⋅arene (1⋅phen⋅arene) (phen=phenazine; arene=toluene, p‐xylene or benzene) have been synthesised from the solution phase in a series of arene solvents and crystallographically characterised. By contrast, analogous syntheses from o‐xylene and m‐xylene as the solvent yield the solvent‐free coordination polymer [Ag4(O2CCF3)4(phen)2] (2). Toluene, p‐xylene and benzene have been successfully used in mixed‐arene syntheses to template the formation of coordination polymers 1⋅phen⋅arene, which incorporate o‐ or m‐xylene. The selectivity of 1⋅phen⋅arene for the arene guests was determined, through pairwise competition experiments, to be p‐xylene>toluene≈benzene>o‐xylene>m‐xylene. The largest selectivity coefficient was determined as 14.2 for p‐xylene:m‐xylene and the smallest was 1.0 for toluene:benzene.

Silver(I) Polymer of Tripodal Amine: Synthesis, Structure and Photoluminescence

Starting from tripodal ligand tris-(methyl-imidazole) amine (1) the colourless complex tris-(methyl-imidazole)aminesilver(I)perchlorate (2) has been synthesized, characterized by IR, UV/Vis and 1HNMR spectroscopic studies and finally solid state structure has been established by single crystal X-ray diffraction studies. A single crystal X-ray structure determination revealed the formation of coordination polymer. The crystal structure possesses weak d10–d10 ‘argentophilic’ interactions with Ag–Ag separation (3.348 A). The weak luminance at 375 nm may be this Ag–Ag weak interaction. The molecule possesses weak H-bonding with C2 and C4H of imidazole with oxygen of ClO4.

Copper(II), zinc(II) and nickel(II) coordination polymers using bidentate hyroxyphenyl-tetrazolyl ligand

Herein, we report various metal complexes containing {5-(2-hydroxy)phenyl}tetrazolyl (H2hptz) which form various high-dimensional coordination polymers in a crystalline solid state. A reaction of metal perchlorates, 2,2′-bipyrimidine (bpym), KOH, and tetrazolyl ligand (H2hptz) in H2O or N,N-dimethylmethanamide (DMF) yield 2-D [Cu(hptz)(bpym)]n (1), [Zn(hptz)(bpym)]n (2) and 3-D [K(bpym)2Ni(hptz)(H2O)2.5(OH)]n (3) coordination complexes with a high synthetic yield. These compounds were characterized by elemental analysis, IR spectroscopy, and single crystal X-ray crystallography. Cu2+ and Zn2+ complexes showed 2-D infinite network systems, forming coordination polymers (caused by extra bonding between metal ions and tetrazolyl). The environment of the Cu and Zn metal centers are quite similar; however, Zn2+ complexes have significantly shorter Zn–O bond distances with the hydroxyl group of other [Zn(hptz)(bpym)] unit. The Ni2+ complex 3 showed a completely different composition from that of complexes 1–2. Complex 3 showed 3-D structure based on octahedral units, which were composed of two hptz2− and one bpym ligands around the Ni center. These units infinitely repeat with K ions and water molecules along the a-, b-, and c-axes of the unit cells. Their magnetic behaviors were also studied.

Coordination Polymers of M2L2 Macrocycles and M3L2 Podands Containing Tris (pyridyl) Tripodal Amide: Anion Bridging, Ag⋅⋅⋅Ag Interactions and Solid‐State Luminescence

AbstractThe tris‐pyridyl‐tripodal amide ligand (L), which is synthesized by the condensation of tris(2‐aminoethyl)amine (tren) and isonicotinic acid, was shown to form coordination polymers (CPs) with AgX (X= CF3SO3−, BF4− and ClO4−), HgX2 (X= Cl− and Br−) and CdI2. The crystal structure analysis of {[Ag3L2(CF3SO3)3].MeOH}n, 1, {[AgL](BF4).2H2O}n, 2, {[AgL](ClO4).MeOH.H2O}n, 3, {[Hg2Cl4L].H2O}n, 4, [Hg3Br6L2], 5, [Cd2I4L2.(H2O)2], 6, reveals that they are constituted by M2L2 macrocycles and M3L2 podand structures. The complexes 1–4 were found to form 1D‐coordination polymers by linking M2L2 units via anions. Complexes 5 and 6 found to contain capsule (podand) like structure with M3L2 and M2L2 H2O compositions in 5 and 6 respectively. The complex 1 contains Ag⋅⋅⋅Ag interactions within the Ag2L2 macrocycle whereas the complex 2 and 3 contains such argentophilic interactions between the macrocycles. These 1D‐CPs linked further via amide‐to‐amide and amide‐to‐water hydrogen bonding to form higher dimensional networks. In these complexes the ligand (L) found to exhibit conformational variations which defines the overall network geometries. The ligand (L) and the coordination complexes 1–6 were found to exhibit intense luminescence emissions upon exciting at a wavelength of 310 nm.

Recovery of metals from simulant spent lithium-ion battery as organophosphonate coordination polymers in aqueous media

An innovative approach is proposed for the recycling of metals from a simulant lithium-ion battery (LIBs) waste aqueous solution. Phosphonate organic linkers are introduced as precipitating agents to selectively react with the metals to form coordination polymers from an aqueous solution containing Ni, Mn and Co in a hydrothermal process. The supernatant is analyzed by ICP-AES to quantify the efficiency and the selectivity of the precipitation and the materials are characterized by Scanning Electron Microscopy (SEM), Powder X-Ray Diffraction (PXRD), Thermogravimetric Analyses (TGA) and nitrogen gas sorption (BET). Conditions have been achieved to selectively precipitate Manganese or Manganese/Cobalt from this solution with a high efficiency. This work describes a novel method to obtain potentially valuable coordination polymers from a waste metal solution that can be generalized on any waste solution.

New flexible bis-pyridyltetrazole ligands for use in coordination polymer formation: Synthesis and structural studies

In this work, we have further developed the pyridyl-tetrazole ligand systems into bis-tetrazole systems, where the linker introduced to join these moieties was a flexible (CH2)3 backbone, with the intention of forming frameworks with potential voids. We synthesised three different pyridyl-tetrazoles linked via this flexible backbone through the pyridine ring and not the tetrazole rings. Two crystals of ligand 1 were solved by X-ray crystallography and showed interesting polymorphism. No crystals of the copper(II) complexes of the ligands could be obtained, suggesting that they were polymeric in nature.

Silver(I) coordination polymers assembled from flexible cyclotriphosphazene ligand: structures, topologies and investigation of the counteranion effects.

The reactions of a flexible ligand hexakis(3-pyridyloxy)cyclotriphosphazene (HPCP) with a variety of silver(I) salts (AgX; X = NO3(-), PF6(-), ClO4(-), CH3PhSO3(-), BF4(-) and CF3SO3(-)) afforded six silver(I) coordination polymers, namely {[Ag2(HPCP)]·(NO3)2·H2O}n (1), {[Ag2(HPCP)(CH3CN)]·(PF6)2}n (2), {[Ag2(HPCP)(CH3CN)]·(ClO4)2}n (3), [Ag3(HPCP)(CH3PhSO3)3]n (4), [Ag2(HPCP)(CH3CN)(BF4)2]n (5) and {[Ag(HPCP)]·(CF3SO3)}n (6). All of the isolated crystalline compounds were structurally determined by X-ray crystallography. Changing the counteranions in the reactions, which were conducted under similar conditions of M/L ratio (1:1), temperature and solvent, resulted in structures with different types of topologies. In complexes (1)-(6), the ligand HPCP shows different coordination modes with Ag(I) ions giving two-dimensional layered structures and three-dimensional frameworks with different topologies. Complex (1) displays a new three-dimensional framework adopting a (3,3,6)-connected 3-nodal net with point symbol {4.6(2)}2{4(2).6(10).8(3)}. Complexes (2) and (3) are isomorphous and have a two-dimensional layered structure showing the same 3,6L60 topology with point symbol {4.2(6)}2{4(8).6(6).8}. Complex (4) is a two-dimensional structure incorporating short Ag...Ag argentophilic interactions and has a uninodal 4-connected sql/Shubnikov tetragonal plane net with {4(4).6(2)} topology. Complex (5) exhibits a novel three-dimensional framework and more suprisingly contains twofold interpenetrated honeycomb-like networks, in which the single net has a trinodal (2,3,5)-connected 3-nodal net with point symbol {6(3).8(6).12}{6(3)}{8}. Complex (6) crystallizes in a trigonal crystal system with the space group R\bar 3 and possesses a three-dimensional polymeric structure showing a binodal (4,6)-connected fsh net with the point symbol (4(3).6(3))2.(4(6).6(6).8(3)). The effect of the counteranions on the formation of coordination polymers is discussed in this study.

Cerium(IV) Oxidations Of Sulfated Polysaccharides In Aqueous Perchlorate Solutions: A Kinetic And Mechanistic Approach

The kinetics of oxidation of iota- and lambda-carrageenans as sulfated polysaccharides by cerium(IV) was studied spectrophotometrically in aqueous perchlorate solutions at a fixed ionic strength of 2.0 mol dm -3 and a temperature of 25 o C. The reactions showed a first order dependence on (Ce IV ), whereas the orders with respect to each carrageenan concentration were less than unity. The reactions exhibited fractional-first order kinetics with respect to (H + ). Increasing ionic strength increased the oxidation rates. The oxidation products of carrageenans were characterized by elemental analysis and IR spectra as their diketo-acid derivatives. The oxidation products were found to have high tendencies to form coordination polymer complexes with some metal cations such as Ba II , Cd II , Pb II and Ag I . Kinetic evidences for the formation of 1:1 complexes were revealed. Plausible mechanistic scheme for cerium(IV) oxidations of carrageenans were proposed. The activation parameters with respect to the slow step of the reactions were evaluated and discussed. The rate laws have been derived and the reaction constants involved in the different steps of the mechanisms were calculated. The activation parameters associated with the rate- determining step of the mechanism along with thermodynamic quantities of the equilibrium constants were computed and discussed. Copyright © 2016 VBRI Press.

Coordination polymers from main group ring compounds

GRAPHICAL ABSTRACT

Mechanochemical Synthesis of 3d Transition-Metal-1,2,4-Triazole Complexes as Precursors for Microwave-Assisted and Thermal Conversion to Coordination Polymers with a High Influence on the Dielectric Properties.

The complexes [MCl2 (TzH)4] (M=Mn (1), Fe (2); TzH=1,2,4-1H-triazole) and [ZnCl2 (TzH)2] (3) have been obtained by mechanochemical reactions of the corresponding divalent metal chloride and 1,2,4-1H-triazole. They were successfully used as precursors for the formation of coordination polymers either by a microwave-assisted reaction or by thermal conversion. For manganese, the conversion directly yielded 1∞ [MnCl2 TzH] (4), whereas for the iron-containing precursor, 1∞ [FeCl2 TzH] (6), was formed via the intermediate coordination polymer 1∞ [FeCl(TzH)2]Cl (5). For cobalt, the isotypic polymer 1∞ [CoCl(TzH)2]Cl (7) was obtained, but exclusively by a microwave-induced reaction directly from CoCl2 . The crystal structures were resolved from single crystals and powders. The dielectric properties were determined and revealed large differences in permittivity between the precursor complexes and the rigid chain-like coordination polymers. Whereas the monomeric complexes exhibit very different dielectric behaviour, depending on the transition metal, from "low-k" to "high-k" with the permittivity ranging from 4.3 to >100 for frequencies of up to 1000 Hz, the coordination polymers and complexes with strong intermolecular interactions are all close to "low-k" materials with very low dielectric constants up to 50 °C. Therefore, the conversion procedures can be used to deliberately influence the dielectric properties from complex to polymer and for different 3d transition-metal ions.

First Report on Crystal Engineering of Hg(II) Halides with Fully Substituted 3,4-Pyridinedicarboxamides: Generation of Two-Dimensional Coordination Polymers and Linear Zig-Zag Chains of Mercury Metal Ions

Two new ligands N,N,N′,N′-tetraisopropyl/tetraisobutyl-3,4-pyridinedicarboxamide L1–L2 and six of their Hg(II)X2 complexes (where X = Cl–, Br–, and I–) have been synthesized and characterized. Single crystal X-ray structures of three complexes of L1 (1–3) with HgCl2/Br2/I2 and two complexes of L2 (4–5) with HgCl2/Br2 show that these are two-dimensional (2D) coordination polymers, with three different (one new) topologies and five coordinated Hg(II) ions, in square pyramidal coordination. Complex 6, of L2 with HgI2, is a dimer with a four coordinated tetrahedral Hg(II) ion. The ligands behave as two- or three-connecting linkers for forming coordination polymers (CPs) but only one-connector in the dimer. Various types of hydrogen bonding and other noncovalent interactions have been calculated, analyzed, and discussed for all the complexes and the ligand L1. Although the large size and soft character of iodide coupled with the steric effects of large isobutyl groups are mainly responsible for a change in the...

Separation of neodymium and dysprosium by forming coordination polymers

Abstract The increasing demand for neodymium (Nd)-magnets containing dysprosium (Dy) has necessitated the recovery of Nd and Dy from magnet scraps to ensure their supply. Thus, it is required to develop an environmentally friendly method for separating Nd3+ and Dy3+. Herein, we show that the formation of coordination polymers based on di(2-ethylhexyl) phosphoric acid (HDEHP) enables the fractional precipitation of Nd and Dy in ethanol–water solution because the solubility of the Dy coordination polymer is significantly lower than that of the Nd coordination polymer owing to coordination preference. The separation performance was found to be better in HNO3 than in HCl media. The characterization of the solid and liquid phases obtained from the Nd3+ and Dy3+ precipitation systems suggested that Nd3+– NO 3 - association inhibits the formation of Nd coordination polymers. The partial replacement of dehp− in the Nd coordination polymer by NO 3 - may weaken the binding strength of the framework, resulting in higher solubility. In addition, the formation of Nd(NO3)2+ in solution may shift the equilibrium against the precipitation reaction. However, when Nd3+ and Dy3+ coexist, Nd3+ precipitation is not accompanied by the replacement of dehp− by NO 3 - in the solid phase. This may be attributed to the incorporation of Nd3+ into the framework of the Dy coordination polymer. We conclude that the formation of Nd(NO3)2+ shifts the solution equilibrium and plays an important role in enhancing the separation performance in HNO3 than in HCl when Nd3+ and Dy3+ coexist.

Thermostability and photophysical properties of mixed-ligand carboxylate/benzimidazole Zn(II)-coordination polymers

The reaction between Zn(NO3)2·6H2O or Zn(CH3COO)2·2H2O and isophthalic acid (1,3-H2bdc) in the presence of benzimidazole (Hbzim) in dimethylformamide (DMF)/ethanol (EtOH)/H2O solvent mixture at room temperature yielded two structurally different coordination polymers: [Zn2(1,3-bdc)2(Hbzim)2] (1) and [Zn2(1,3-bdc)(bzim)2] (2). (1) is a 2D-layered framework with a molecule of benzimidazole coordinated to the Zn center, whereas (2) is a 3D framework with benzimidazolate species acting as a co-ligand and bridging two Zn(II) ions. Reactions performed at 90 °C led to the formation of coordination polymers structurally similar to (2), independently of the Zn(II) source used. In the absence of benzimidazole, the reaction between ZnAc2.2H2O and 1,3-H2bdc at 90 °C resulted in the formation of (3), a 3D coordination polymer Zn(HCOO)3(Me2NH2+). It was observed that the thermostability and the photophysical properties of (1) and (2) are strongly dependent on the coordination modes and packing of benzimidazole in the solid state. These materials present photoluminescence in the wide range of the spectrum, from UV to IR. A full understanding of a physical process occurring in these intriguing systems, including complete energy level diagrams with possible transitions were provided.

Zinc complexes of pyridyl-tetrazole derivatives – Highly fluorescent materials

The ligands L1Et and L2Et were synthesised and hydrolysed in situ to form carboxylates, utilising methods previously established, for the purpose of investigating the ability of carboxylate functionalised pyridyl tetrazoles to form coordination polymers. These ligands were then reacted with ZnCl2 and displayed interesting coordination chemistry as the position of the alkylation site appeared to have some effect on the coordination sphere of the metal centre. Solid state fluorescence spectroscopy was carried out on Zn(II) compounds and their corresponding free acid ligands at room temperature. The crystal structure of L1Et was also solved and showed the co-planarity of the two rings.