Path selection for conformational interconversions in [2]catenanes

[strucure: see text]The conformational interconversions of several [2]catenanes containing a dibenzo-34-crown-10 ether (BPP34C10) interlocked with rings containing two 4,4'-dipyridyls tethered by different aryl spacers have been studied. Blocking groups on the tethers enabled the two pathways for circumrotation of the BPP34C10 to be open or blocked. The activation barrier for migration along the open tethers varied from 11 to 13 kcal/mol. This study demonstrates an ability to select the pathway for conformational interconversions in [2]catenanes.     

Translational isomerizations in [2]catenanes with unsymmetrically substituted resorcinol-based tethers

The translational isomerizations of nine [2]catenanes (2-10) containing an electron-rich dibenzo-34-crown-10 ether (BPP34C10) interlocked with rings containing an unsymmetric 4-substituted (chloro, ethyl, or hexyl) resorcinol-based tether linking two electron-deficient dipyridyl groups have been studied by variable temperature (VT) 1H NMR spectroscopy. The second symmetric tether between the dipyridyl groups was a large 5-(4-tert-butylphenyl)-1,3-xylyl (catenanes 2-4), a narrower 1,3-xylyl (catenanes 5-7), or a narrow 1,4-xylyl (catenanes 8-10) group. The presence of the unsymmetrically placed substituent on the resorcinol tether substantially affected the binding energy of the BPP34C10 ring when pi-stacked over either of the dipyridyl groups; the equilibrium constant between the bistable states was found to range from 1.5 to 3.5. The origin of these energy differences is postulated to stem from an unsymmetric twisting of the resorcinol tether to minimize interaction between the 4-substituent and the ethoxy group at the 3-position. The activation barriers for passage over the 4-substituted resorcinol-based tether were 12.5, 13, and 15 kcal/mol for the chloro, ethyl, and hexyl substituents, respectively.     

Dynamic chirality: keen selection in the face of stereochemical diversity in mechanically bonded compounds

The template-directed syntheses, employing bisparaphenylene-[34]crown-10 (BPP34C10), 1,5-dinaphthoparaphenylene-[36]crown-10 (1/5NPPP36C10), and 1,5-dinaphtho-[38]crown-10 (1/5DNP38C10) as templates, of three [2]catenanes, whereby one of the two bipyridinium units in cyclobis(paraquat-p-phenylene) is replaced by a bipicolinium unit, are described. The crude reaction mixtures comprising the [2]catenanes all contain slightly more of the homologous [3]catenanes, wherein a "dimeric" octacationic cyclophane has the crown ether macrocycles encircling the alternating bipyridinium units with the bipicolinium units completely unfettered. X-ray crystallography, performed on all three [2]catenanes and two of the three [3]catenanes reveals co-conformational and stereochemical preferences that are stark and pronounced. Both the [3]catenanes crystallize as mixtures of diastereoisomers on account of the axial chirality associated with the picolinium units in the solid state. Dynamic (1)H NMR spectroscopy is employed to probe in solution the relative energy barriers for rotations by the phenylene and pyridinium rings in the tetracationic cyclophane component of the [2]catenanes. Where there are co-conformational changes that are stereochemically "allowed", crown ether circumrotation and rocking processes are also investigated for the relative rates of their occurrence. The outcome is one whereby the three [2]catenanes containing BPP34C10, 1/5NPPP36C10, and 1/5DNP38C10 exist as one major enantiomeric pair of diastereoisomers amongst two, four, and eight diastereoisomeric pairs of enantiomers, respectively. The diastereoisomerism is a consequence of the presence of axial chirality together with helical and/or planar chirality in the same interlocked molecule. These [2]catenanes constitute a rich reserve of new stereochemical types that might be tapped for their switching and mechanical properties.     

A photochemically driven molecular-level abacus

A molecular-level abacus-like system driven by light inputs has been designed in the form of a [2]rotaxane, comprising the pi-electron-donating macrocyclic polyether bis-p-phenylene-34-crown-10 (BPP34C10) and a dumbbell-shaped component that contains 1) a Ru(II) polypyridine complex as one of its stoppers in the form of a photoactive unit, 2) a p-terphenyl-type ring system as a rigid spacer, 3) a 4,4'-bipyridinium unit and a 3,3'-dimethyl-4,4'-bipyridinium unit as pi-electron-accepting stations, and 4) a tetraarylmethane group as the second stopper. The synthesis of the [2]rotaxane was accomplished in four successive stages. First of all, the dumbbell-shaped component of the [2]rotaxane was constructed by using conventional synthetic methodology to make 1) the so-called "west-side" comprised of the Ru(II) polypyridine complex linked by a bismethylene spacer to the p-terphenyl-type ring system terminated by a benzylic bromomethyl function and 2) the so-called "east-side" comprised of the tetraarylmethane group, attached by a polyether linkage to the bipyridinium unit, itself joined in turn by a trismethylene spacer to an incipient 3,3'-dimethyl-4,4'-bipyridinium unit. Next, 3) the "west-side" and "east-side" were fused together by means of an alkylation to give the dumbbell-shaped compound, which was 4) finally subjected to a thermodynamically driven slippage reaction, with BPP34C10 as the ring, to afford the [2]rotaxane. The structure of this interlocked molecular compound was characterized by mass spectrometry and NMR spectroscopy, which also established, along with cyclic voltammetry, the co-conformational behavior of the molecular shuttle. The stable translational isomer is the one in which the BPP34C10 component encircles the 4,4'-bipyridinium unit, in keeping with the fact that this station is a better pi-electron acceptor than the other station. This observation raises the question- can the BPP34C10 macrocycle be made to shuttle between the two stations by a sequence of photoinduced electron transfer processes? In order to find an answer to this question, the electrochemical, photophysical, and photochemical (under continuous and pulsed excitation) properties of the [2]rotaxane, its dumbbell-shaped component, and some model compounds containing electro- and photoactive units have been investigated. In an attempt to obtain the photoinduced abacus-like movement of the BPP34C10 macrocycle between the two stations, two strategies have been employed-one was based fully on processes that involved only the rotaxane components (intramolecular mechanism), while the other one required the help of external reactants (sacrificial mechanism). Both mechanisms imply a sequence of four steps (destabilization of the stable translational isomer, macrocyclic ring displacement, electronic reset, and nuclear reset) that have to compete with energy-wasteful steps. The results have demonstrated that photochemically driven switching can be performed successfully by the sacrificial mechanism, whereas, in the case of the intramolecular mechanism, it would appear that the electronic reset of the system is faster than the ring displacement.     

The Nature of the [TTF].+⋅⋅⋅[TTF].+ Interactions in the [TTF]22+ Dimers Embedded in Charged [3]Catenanes: Room‐Temperature Multicenter Long Bonds

The properties of tetrathiafulvalene dimers ([TTF]₂²⁺) and the functionalized ring‐shaped bispropargyl (BPP)‐functionalized TTF dimers, [BPP–TTF]₂²⁺, found at room temperature in charged [3]catenanes, were evaluated by M06L calculations. The results showed that their isolated [TTF]₂²⁺ and [BPP–TTF]₂²⁺ dimers are energetically unstable towards dissociation. When enclosed in the 4⁺‐charged central cyclophane ring of charged [3]catenanes (CBPQT⁴⁺), [TTF]₂²⁺ and [BPP–TTF]₂²⁺ dimers are also energetically unstable with respect to leaving the CBPQT⁴⁺ ring; since the barrier for the exiting process is only about 3 kcal mol^?1, that is, within the reach of thermal energies at room temperature (neutral [TTF]₂⁰ dimers are stable within the CBPQT⁴⁺ ring). However, the [BPP–TTF]₂²⁺ dimers in charged [3]catenanes cannot exit, because this would imply breaking the covalent bonds of the BPP–TTF⁺ macrocycle. Finally, it was shown that the [TTF]₂²⁺, [BPP–TTF]₂²⁺ dimers, and charged [3]catenanes are energetically stable in solution and in crystals of their salts, in the first case due to the interactions with the solvent, and in the second case mostly due to cation–anion interactions. In these environmental conditions at room temperature the TTF units of the [BPP–TTF]₂²⁺ dimers make short contacts, thus allowing their SOMO orbitals to overlap: a room‐temperature multicenter long bond is formed, similar to those previously found in other [TTF]₂²⁺ salts and their solutions.     

Highly diastereoselective type-I IMDA reaction forming medium-sized macrolactones

[Structure: see text] Intramolecular Diels-Alder (IMDA) reactions of 2-pyrones containing an alkyne tether occur with remarkably high diastereofacial selectivity to provide medium-sized macrolactones. Because of the strain in the alkyne-tethered macrocyclic system, a single methyl group in the tether provides sufficient conformational bias to generate medium-sized macrocycles with unusually high selectivity.     

Synthesis and Characterization of Two [2]Catenanes Based on Phenylene-diacetylene Crown Ethers

Two charged donor-acceptor [2]catenanes were synthesized by the cyclization of the π-acceptor 1,1 ＇-[1, 4-phenylenebis-（methylene）]bis（4,4＇-bipyridinium）bis（hexafluorophosphate）（2） and bis（bromomethyl）benzene with the templates of the π-donor phenylene-diacetylene crown ethers la and lb in the yields of 31%, 35%, respectively. The mass spectra, 1H and 13C NMR spectra confirm that the phenylene-diacetylene crown ethers, with a nonaromatic ~r-system, successfully template the cyclization of CBPQT4＋. The X-ray crystallography analysis shows that the CBPQT4＋ ring encircles around the phenylene motif of the crown ethers, leaving the 1,3-butadiyne fragment out of the cavity of CBPQT4＋ ring. The mechanically interlocked structure of [2]catenanes was stabilized by the cooperative effects of π-stacking and hydrogen bonding interactions.     

When Self‐Assembly Fails: Stepwise Metal‐Directed Synthesis of [2]Catenanes

On the attempted synthesis of a series of homo‐ and heterotrimetallic [2]catenanes by the self‐assembly of a 2‐(pyridin‐4‐ylmethyl)‐2,7‐diazapyrenium ligand, (ethylenediamine)palladium(II) or platinum(II) nitrate, and a dioxoaryl bis(N‐monoalkyl‐4,4′‐bipyridinium) salt as building blocks, both the one‐pot direct self‐assembly of the components and the so called “magic ring” approach fail to produce the expected trinuclear [2]catenanes under thermodynamically driven conditions. However, one of the target supramolecules is obtained by following a stepwise protocol, consisting of the threading of a dinuclear Pt^II metallacycle and the dioxoaryl bis(N‐monoalkyl‐4,4′‐bipyridinium) axle, followed by kinetically controlled Pt^II‐directed cyclization of the corresponding pseudorotaxane.     

Spectroscopic and Electrochemical Properties of Catenanes Containing the 2,7-Diazapyrenium Unit

Abstract

The spectroscopic and electrochemical properties of two cyclophanes containing one and, respectively, two 2,7-diazapyrenium electron-acceptor units, and of their [2]catenanes with macrocycles containing two dioxybenzene or dioxynaphthalene electrondonor units have been investigated. The absorption spectra of the catenanes show weak and broad bands in the visible region, assigned to charge-transfer (CT) interactions. The very strong and structured fluorescence (298 K) and the structured fluorescence and phosphorescence (77 K) of the diazapyrenium unit are maintained in the two cyclophanes, but they are no longer present in the [2]catenanes, presumably because of a quenching process caused by the lower energy CT excited states. Each diazapyrenium unit undergoes two distinct reduction processes - only the first one of which is fully reversible - that are hardly affected at all when the diazapyrenium units are incorporated in a cyclophane. In the [2]catenanes, the CT interaction displaces the reduction processes of the diazapyrenium units toward more negative potentials. The results obtained for the diazapyrenium and previously investigated 4,4′-bipyridinium salts, selected cyclophane derivatives, and some [2]catenanes obtained by interlocking the cyclophanes with macrocycles containing two dioxyaromatic electron-donor units are compared and discussed.     

Acid/base controllable self-complexes that mimic flapping butterfly

A new type of self-complexed bis-crown ether containing two bis(p-phenylene)-34-crown-10 (BPP34C10) ether rings and two secondary ammoniums has been synthesized and characterized. The formation of these bis-self-complexes has been identified by NMR spectroscopy, mass spectrometry and X-ray analysis. The acid/base controlled movement of these bis–crown ethers can mimic a flapping butterfly.     

The nature of the [TTF]˙+···[TTF]˙+ interactions in the [TTF]2(2+) dimers embedded in charged [3]catenanes: room-temperature multicenter long bonds

The properties of tetrathiafulvalene dimers ([TTF](2)(2+)) and the functionalized ring-shaped bispropargyl (BPP)-functionalized TTF dimers, [BPP-TTF](2)(2+), found at room temperature in charged [3]catenanes, were evaluated by M06L calculations. The results showed that their isolated [TTF](2)(2+) and [BPP-TTF](2)(2+) dimers are energetically unstable towards dissociation. When enclosed in the 4(+)-charged central cyclophane ring of charged [3]catenanes (CBPQT(4+)), [TTF](2)(2+) and [BPP-TTF](2)(2+) dimers are also energetically unstable with respect to leaving the CBPQT(4+) ring; since the barrier for the exiting process is only about 3 kcal mol(-1), that is, within the reach of thermal energies at room temperature (neutral [TTF](2)(0) dimers are stable within the CBPQT(4+) ring). However, the [BPP-TTF](2)(2+) dimers in charged [3]catenanes cannot exit, because this would imply breaking the covalent bonds of the BPP-TTF(+) macrocycle. Finally, it was shown that the [TTF](2)(2+), [BPP-TTF](2)(2+) dimers, and charged [3]catenanes are energetically stable in solution and in crystals of their salts, in the first case due to the interactions with the solvent, and in the second case mostly due to cation-anion interactions. In these environmental conditions at room temperature the TTF units of the [BPP-TTF](2)(2+) dimers make short contacts, thus allowing their SOMO orbitals to overlap: a room-temperature multicenter long bond is formed, similar to those previously found in other [TTF](2)(2+) salts and their solutions.     

Synthesis of [3]catenanes based on metal-directed self-assembly and pi-donor/pi-acceptor interactions

A metal-directed self-assembly of [3]catenanes in combination with pi-pi interactions was investigated. Ligands based on 4,4'-bipyridinium or 2,7-diazapyrenium were used in conjunction with dioxoaryl cyclophanes (4-6) and trans-PdCl2(CH3CN)2. The [3]catenanes show a dinuclear palladium 46-membered metallomacrocycle interlocked by two pi-complementary dioxoaryl macrocycles. [structure: see text].     

Surface‐Bound Cucurbit[8]uril Catenanes on Magnetic Nanoparticles Exhibiting Molecular Recognition

We demonstrate the preparation of surface‐bound cucurbit[8]uril (CB[8]) catenanes on silica nanoparticles (NPs), where CB[8] was employed as a tethered supramolecular “handcuff” to selectively capture target guest molecules. In this catenane, CB[8] was threaded onto a methyl viologen (MV²⁺) axle and immobilized onto silica NPs. The formation of CB[8] catenanes on NPs were confirmed by UV/Vis titration experiments and lithographic characterization, demonstrating a high density of CB[8] on the silica NPs surface, 0.56 nm^?2. This CB[8] catenane system exhibits specific molecular recognition towards certain aromatic molecules such as perylene bis(diimide), naphthol and aromatic amino acids, and thus it can act as a nanoscale molecular receptor for target guests. Furthermore, we also demonstrate its use as an efficient and recyclable nano‐platform for peptide separation. By embedding magnetic NPs inside silica NPs, separation could be achieved by simply applying an external magnetic field. Moreover, the peptides captured by the catenanes could be released by reversible single‐electron reduction of MV²⁺. The entire process demonstrated high recoverability.     

Template effects and kinetic selection in the self-assembly of crown ether cyclobis(paraquat-p-phenylene)

The template effects exerted by bis(p-phenylene)[34]crown-10 (3) and by 1,5-dinaphto[38]crown-10 (4) in the ring-closure reaction of the trication 2(3+) to yield the [2]catenanes 7(4+) and 8(4+) have been quantitatively evaluated in acetonitrile at 62 degrees C by UV/visible spectroscopy. The rate of ring closure of the trication 2(3+) dramatically increases in the presence of the templates 3 and 4, up to approximately 230 times at [3] approximately equals 0.1 molL(-1), and up to approximately 1,900 times at [4] approximately equals 0.01 molL(-1). The outcome of kinetic selection experiments, in which the two crown ethers compete for the incorporation into the catenane structure, has been discussed in the light of the obtained results. It has been shown that the product ratio of catenanes obeys the Curtin-Hammett principle only if the concentrations of the templates are equal and much greater than that of the substrate. Analysis of the rate profiles has shown that the 1,5-dioxynaphthalene unit, present in the template 4, has a greater affinity than the 1,4-dioxybenzene unit, present in the template 3, for the electron-deficient pyridinium rings present in both the transition-state and substrate structures. Ab initio calculations at the 3-21G and 6-31G(d) levels of theory indicate that the greater affinity of the 1,5-dioxynaphthalene unit cannot be explained on the basis of greater pi-pi stacking and [C-H...pi] interactions, but rather on the basis of the model of apolar complexation in which the solvent plays a major role.     

Double bonds in motion: bis(oxazolinylmethyl)pyrroles and their metal-induced planarization to a new class of rigid chiral C2-symmetric complexes

The synthesis of a new class of chiral C(2)-symmetric tridentate N-donor ligands, a series of 2,5-bis(2-oxazolinylmethyl)pyrroles, was achieved in four steps starting from the known 2,5-bis(trimethylammoniomethyl)pyrrole diiodide (1). Reaction of 1 with NaCN in dimethyl sulfoxide gave 2,5-bis(cyanomethyl)pyrrole (2) cleanly, which was then cyclized with amino alcohols to give the 2,5-bis(2-oxazolinylmethyl)pyrroles 3 a-c (3 a: bis[2-(4,4'-dimethyl-5-hydrooxazolyl)methyl]pyrrole; 3 b: (S,S)-bis[2-(4-isopropyl-4,5-dihydrooxazolyl)methyl]pyrrole; 3 c: (S,S)-bis[2-(4-tertiobutyl-4,5-dihydrooxazolyl)methyl]pyrrole). Metallation of 3 a-c with one molar equivalent of tBuLi and their subsequent reaction with a stoichiometric amount of [PdCl(2)(cod)] (cod=cyclooctadiene) gave the palladium(II) complexes 4 a-c. Whereas the arrangement of the N-donor atoms in the crystallographically characterized complex 4 a is almost ideally square planar, all three heterocycles in the ligand are twisted out of the coordination plane, leading to a chiral conformation of the complex. Attempts to freeze out these two conformers in solution at 200 K (NMR) failed, and this suggests that the activation barrier for conformational racemization is significantly below 10 kcal mol(-1). The palladium-induced shift of two double bonds as well as the porphyrinogen/porphyrin-type oxidation of the complexes 4 a-c led to the planarization of the 2,5-bis(oxazolinylmethyl)pyrrolide ligands in the palladium(II) complexes 5 a-c, 6 b, and 6 c, and to the formation of rigid chiral C(2)-symmetric systems as shown by X-ray diffraction studies. The formation of the conjugated system of double bonds in this transformation is accompanied by the emergence of an intra-ligand chromophore. This is evident in the absorption spectrum of 6 c which displays an intense band with a maximum at 485 nm attributable to an intra-ligand pi*<--pi transition and a characteristic vibrational progression of nu approximately 1350 cm(-1). Complexes 4 b and 4 c were tested in the catalytic asymmetric Michael addition of ethyl 2-cyanopropionate to methyl vinylketone (catalyst loading: 1 mol %) and were found to give maximum ee values of 43 % (4 b) and 21 % (4 c) at low conversions.     

固相熔融合成含4,4'-联吡啶二阳离子的哑铃型化合物

2-{2-[4-苯基-二(4-特丁基苯基)甲基]苯氧基}乙氧乙醇磺酸酯(2.1 mmol)与4,4'-联吡啶(1.0 mmol)混合, 加热到融化, 搅拌10 min, 用石油醚洗涤反应混合物, 得到定量的哑铃型化合物.     

Calix[4]arene-based bis[2]catenanes: synthesis and chiral resolution

The exclusive formation of hydrogen-bonded dimers between tetraaryl and tetratosylurea calix[4]arenes has been used to prepare a series of ten "bisloop" tetraurea calix[4]arenes 3, in which adjacent phenylurea groups are covalently linked through alpha,omega-dioxyalkane chains. This dimerization with tetratosylurea 2 as template preorganizes the alkenyl residues of tetra(m-alkenyloxyphenyl) ureas 1 and enables their selective connection in high yield (up to 95 %) by olefin metathesis followed by hydrogenation. The "bisloop" calixarenes 3 also exclusively form heterodimers with 1. Thus, in a separated metathesis/hydrogenation sequence, a series of 14 cyclic bis[2]catenanes 4, in which two calix[4]arenes are connected through their wide rims by two pairs of interlocked rings (total size 29 to 41 atoms), were prepared in yields of up to 97 %. Optical resolution of these chiral bis[2]catenanes was studied by HPLC on chiral stationary phases. The single-crystal X-ray structure of one example (4(P,10)) confirmed the interlocking rings and revealed that the hydrogen-bonded dimeric capsule of the calix[4]arene can be "completely" opened.     

Molecular catenation via metal-directed self-assembly and pi-donor/pi-acceptor interactions: efficient one-pot synthesis, characterization, and crystal structures of [3]catenanes based on Pd or Pt dinuclear metallocycles

Dinuclear square metallocycles 3a,b assemble spontaneously when M(en)(OTf)2 (M = Pd, Pt) and a 4,4'-bipyridinium ligand are mixed in acetonitrile. Six new [3]catenanes were prepared in good yields by thermodynamically driven self-assembly reaction of molecular squares 3a,b and pi-complementary dioxoaryl cyclophanes. Single-crystal X-ray analyses of the [3]catenanes revealed the insertion of two aromatic units inside the metallocycle cavity. The structures are stabilized by means of a combination of pi-pi stacking, [C-H...pi] interactions, and [C-H...O] hydrogen bonds. [3]Catenane (DB24C8)2-(3a) showed in solid-state two external DB24C8 rings positioned over the Pd(en) corners, which are held in position by [N-H...O] hydrogen bonds. Furthermore, formation of catenane (DB24C8)2-(3a) can be switched off and on in a controllable manner by successive addition of KPF6 and 18-crown-6.     

Preparation of poly(phosphate ester)s having bisphenol moieties as mesogenic units in the main chain

Poly(phosphate ester)s, PPE 1a–d , were synthesized from polycondensation of methyl phosphorodichloridate (MPDC) with various bisphenols such as 4,4′-biphenol 1a , 4,4′-dihydroxyphenylether 1b , bis(4-hydroxyphenyl)methane 1c , and 3,3′-dimethyl-4,4′-dihy-droxybiphenyl 1d . PPE 2a–d with hexamethylene spacers were also obtained from poly-condensation of MPDC with 4,4′-bis(6-hydroxyhexyloxy)biphenyl 2a , 4,4′-di(6-hydroxyhexyloxy)phenyl ether 2b , bis[4-(6-hydroxyhexyloxy)phenyl]methane 2c , and 3,3′-dimethyl 4,4′-di(6-hydroxyhexyloxy)biphenyl 2d . The degree of crystallinity of PPE 1a–1d without hexamethylene spacer was 3.3–17.6%, whereas PPE 2a and PPE 2b which exhibit mesomorphic behavior were 20.1 and 18.6%, respectively. PPE 2a and PPE 2b show the mesophase at 139.6–195.5°C and 42.4–66.3°C, respectively. PPE 2c and PPE 2d were obtained as rubbery. From pyrolysis of PPE in air the temperature corresponding to 5% weight loss was found to be 322–408°C and 284–291°C for PPE 1 and PPE 2 , respectively. It was also found that PPE 2a was enzymatically degraded by phospholipase C. © 1994 John Wiley & Sons, Inc.     

Backbone‐Directed Self‐Assembly of Interlocked Molecular Cyclic Metalla[3]Catenanes

The efficient backbone‐directed self‐assembly of cyclic metalla[3]catenanes by the combination of tetrachloroperylenediimide (TCPDI)‐based dinuclear rhodium(III) clips and 4,4′‐diazopyridine or 4,4′‐dipyridylethylene ligands is realized in a single‐step strategy. The topology and coordination geometry of the cyclic metalla[3]catenanes are characterized by NMR spectroscopy, ESI‐TOF‐MS spectrometry, UV/Vis‐NIR spectroscopy, and X‐ray diffraction studies. The most remarkable feature of the formed cyclic metalla[3]catenane is that it contains π‐aggregates (ca. 2.6 nm) incorporating six TCPDIs. Further studies revealed that cyclic metalla[3]catenanes can be converted reversibly to their corresponding sodium adducts and precursor building blocks, respectively. This strategy opens the possibility of generating unique supramolecular structures from discrete functional π‐aggregates with precise arrangements.