Abstract
The tetraaminoresorcin[4]arene was synthesized, which otherwise could not be easily synthesized by conventional method, by first introducing the azo group at the upper rim, followed by its reduction with Na2S2O4. Tetraaminoresorcin[4]arene was then condensed with different aromatic aldehydes in ethanol at reflux temperature to obtain nine novel “upper rim” functionalized resorcin[4]arene Schiff-bases. All the nine Schiff-bases were characterized by m.p., elemental analysis, FT-IR, NMR and Mass spectral data.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Introduction
Resorcin[4]arenes (1) are cyclic tetramers readily formed by the acid-catalysed condensation of resorcinol with aldehydes [1]. The most stable crown conformer possesses a bowl-shaped molecular cavity formed by the four resorcinol units [1, 2]. Resorcin[4]arenes deliver a versatile molecular platform for the elaboration of more complicated host system by virtue of its conformational flexibility. This has been widely exploited as a basis for making macrocyclic host molecules in a variety of supramolecular systems, as well as being the basis for their spontaneous adhesion to hydrophilic surfaces and formation of hexameric capsules [3–6]. Their chemistry is mainly focused in (i) synthesis and application of host- guest complexes by inter molecular forces and (ii) construction of novel supramolecular assemblies, dendrimer, nano particles, nanocapsules and metal ion extraction agents [7–11]. Resorcin[4]arenes form a relatively shallow conical cavity that can be extended by suitable substitution and further functionalization. Interest in resorcin[4]arenes has grown rapidly also because of the numerous derivatization that can be created through relatively simple synthetic procedures by substitution at their upper rim, methylene bridges and –OH group at the extra annular position to alter their properties and applications.
Our focus in current the work is to be able to introduce amino functional groups at the upper rim which otherwise could not be achieved by conventional method of Nitration (HNO3/H2SO4) and reduction with common reducing agents like Fe/HCl, Sn/HCl, hydrazine hydrate/raney nickel. In fact Cram et al. [12] also failed to obtain tetra nitro resorcin[4]arene by taking 2-nitro resorcinol and acetaldehyde as the starting materials.
Therefore to obtain amino group at the upper rim, we have first coupled the resorcin[4]arene with diazonium salt of p-aminobenzoic acid and subsequently breaking –N=N– group with sodium dithionite in alkaline medium to get the desired tetra amino product. The tetra amino resorcin[4]arenes was then condensed with different aromatic aldehydes to give corresponding Schiff-bases. Schiff base are versatile coordinating ligands and found to be excellent reagents for biological and industrial purposes [13, 14]. Binding properties of Schiff-bases of resocin[4]arene with various metal ions and guests is yet to be explored.
Experimental
Materials, instruments and methods
All reagents were obtained from Sigma Aldrich or BDH and used without further purification. Resorcin[4]arene was synthesized by earlier reported method [1, 11]. Melting points were taken in a single capillary tube using a Veego melting point apparatus (Model: VMP-DS, India) and were uncorrected. Elemental analysis was done on Perkin Elmer, Series II, 2400 elemental analyzer. IR spectra were recorded on Bruker tensor 27 Infrared spectrophotometer as KBr pellets and expressed in cm−1. 1HNMR spectra were recorded in CDCl3 or DMSO-d6 on a Bruker ARX 500 instrument, using tetramethylsilane as internal standard. Mass Spectra were recorded on JEOL SX 102/DA 6000 mass spectrometer using Xenon/argon (6 KV, 10 mA) as the FAB-gas.
Synthesis
Synthesis of tetra azo resorcin[4]arene (2)
p-amino benzoic acid (1.11 g, 0.0081 M) and 2 mL conc. HCl were added to 15 mL of water and the resulting solution was cooled to 0 °C. Then a solution of NaNO2 (0.6 g, 0.0087 M) in 10 mL of water was added drop wise at such a rate to maintain the temperature below 5 °C. The resulting mixture was slowly added to a solution of resorcin[4]arene (1 g, 0.0018 M) and sodium hydroxide (0.8 g, 0.02 M) in 20 mL of water at 0 °C–5 °C. The reaction mixture was stirred for 2–4 h at the same temperature, and the product was obtained by salting out with sodium chloride. The solid obtained was dissolved in a solution of NaHCO3 and reprecipitated by adding conc. HCl. The mixture was stirred overnight at room temperature, filtered as a red solid and washed with water followed by methanol and dried overnight at 80 °C to yield 72% as a red solid, mp > 280 °C(decompose). 1HNMR (DMSO-d6) δ 1.65 (d, 12H, CH3), 4.67 (q, 4H, bridge CH), 9.78 (s, 12H, OH & COOH), 7.25–7.89 (m, 20H, ArH). IR (KBr) ν/cm−1 3167 (OH), 1698 (C=O), 1517 (N=N), 853. MS-FAB:(M + H)+1137. Elemental analysis for C60H48O16N8 cacld: C, 63.28; H, 4.22; N, 9.86. Found: C, 63.32; H, 4.21; N, 9.79.
Synthesis for tetra amino resorcin[4]arene (3)
The azo compound (2) (0.5 g) was dissolved in a mixture of NaOH (0.5 g) and 25 mL water. The resulting dark red solution was reduced by stirring with Na2S2O4 (2.5 g, 14.4 m mole) for 1 h at 90 °C. The reaction mixture was cooled to room temperature and 20 mL conc. HCl was added. The resulting precipitate were filtered and washed with dil. HCl and methanol. The product was dried at reduced pressure to give 240 mg of 3. The solid was unstable and used in the following reaction without further purification.
Synthesis of resorcin[4]arene Schiff-bases (4)
A mixture of (3) (1 g, 1.655 m mol) and different aromatic aldehydes (8.275 m mol) in 50 mL of ethanol was refluxed for 3–5 h at the boiling temperature. The precipitates obtained were filtered, washed with hot water and recrystallized in ethanol. The resulting solid was dried under vacuum to give yields of 60%–70% for different aldehydes 4a-i.
Benzaldehyde based resorcin[4]arenes Schiff base (4a): Yield 66%; M.P. 135 °C; color: yellow. 1HNMR (CDCl3): δ = 1.65(d, 12H CH3), 4.63(q, 4H bridge CH), 9.49(s, 8H –OH), 8.82(s, 4H –N=CH), 7.3–7.9(s, 24H ArH). IR (KBr): ν = 3,426 cm−1 (–OH), 1,634 cm−1 (–N=CH). MS-FAB:(M + H)+958. Anal.calcd.for C60H52N4O8: C, 75.30; H, 5.48; N, 5.85. Found: C, 74.98; H, 5.44; N, 5.60.
Salisaldehyde based resorcin[4]arenes Schiff base (4b): Yield 68%; M.P. 170 °C; color: yellow. 1HNMR (CDCl3): δ = 1.65(d, 12H CH3), 4.63(q, 4H bridge CH), 9.50(s, 12H –OH), 8.82(s, 4H –N=CH) 7.3–7.9(s, 20H ArH). IR (KBr): ν = 3,403 cm−1 (–OH), 1,604 cm−1 (–N=CH). MS-FAB:(M + H)+1022. Anal.calcd.for C60H52N4O12: C, 70.58; H, 5.13; N, 5.49. Found: C, 70.49; H, 5.10; N, 5.55.
Naphthaldehyde based resorcin[4]arenes Schiff base (4c): Yield 70%; M.P. 145 °C; color: yellow. 1HNMR (CDCl3): δ = 1.64(d, 12H CH3), 4.64(q, 4H bridge CH), 9.58 (s, 8H –OH), 8.88(s, 4H –N=CH) 7.3–7.9(s, 32H ArH). IR (KBr): ν = 3,346 cm−1 (–OH), 1,614 cm−1 (–N=CH). MS-FAB:(M + H)+1158. Anal.calcd.for C76H60N4O8: C, 78.87; H, 5.23; N, 4.84. Found: C, 78.77; H, 5.10; N, 4.72.
2-hydroxy naphthaldehyde based resorcin[4]arenes Schiff base (4d): Yield 64%; M.P. 175 °C (decomp.); color: brown. 1HNMR (CDCl3): δ = 1.64(d, 12H CH3), 4.62(q, 4H bridge CH), 9.50(s, 12H –OH), 8.82(s, 4H –N=CH) 7.3–7.9(s, 28H ArH). IR (KBr): ν = 3,473 cm−1 (–OH), 1,610 cm−1 (–N=CH). MS-FAB:(M + H)+1222. Anal.calcd.for C76H60N4O12 C, 74.74; H, 4.95; N, 4.59. Found: C, 74.98; H, 4.94; N, 4.60.
Cinnamaldehyde based resorcin[4]arenes Schiff base (4e): Yield 62%; M.P. 160 °C; color: creamish. 1HNMR (CDCl3): δ = 1.65(d, 12H CH3), 4.63(q, 4H bridge CH), 4.6(d, 8H –CH=CH–), 9.61(s, 8H –OH),8.89(s, 4H –N=CH), 7.3–7.9(s, 24H ArH). IR (KBr): ν = 3,463 cm−1 (–OH), 1,617 cm−1 (–N=CH), 931(CH=CH). MS-FAB:(M + H)+1062. Anal.calcd.for C68H60N4O8: C, 76.96; H, 5.70; N, 5.28. Found: C, 76.90; H, 5.69; N, 5.18.
o-vanillin based resorcin[4]arenes Schiff base (4f): Yield 62%; M.P. 150 °C; color: light brown. 1H NMR (CDCl3): δ = 1.65(d, 12H CH3), 3.72(s, 12H OCH3), 4.63(q, 4H bridge CH), 9.60(s, 12 H –OH), 8.87(s, 4H –N=CH) 7.3–7.9(s, 16H ArH). IR (KBr): ν = 3,427 cm−1 (–OH), 1,623 cm−1 (–N=CH). MS-FAB:(M + H)+1142. Anal.calcd.for C64H60N4O16: C, 67.36; H, 5.26; N, 4.91. Found: C, 67.42; H, 5.50; N, 4.92.
Vanillin based resorcin[4]arenes Schiff base (4g): Yield 60%; M.P. 125 °C; color: yellow. 1HNMR (CDCl3): δ = 1.65(d, 12H CH3), 3.70(s, 12H OCH3), 4.65(q, 4H bridge CH), 9.71(s, 12H –OH), 8.82(s, 4H –N=CH), 7.4–7.9(s, 16H ArH). IR (KBr): ν = 3,423 cm−1 (–OH), 1,617 cm−1 (–N=CH). MS-FAB:(M + H)+1142. Anal.calcd. for C64H60N4O16: C, 67.36; H, 5.26; N, 4.91. Found: C, 67.41; H, 5.50; N, 4.90.
p-hydroxy benzaldehyde based resorcin[4]arenes Schiff base (4h): Yield 66%; M.P. 127 °C; color: yellow. 1HNMR (CDCl3): δ = 1.64(d, 12H CH3), 4.64(q, 4H bridge CH), 9.68(s, 12H –OH), 8.89(s, 4H –N=CH), 7.3–7.9(s, 20H ArH). IR (KBr): ν = 3,363 cm−1 (–OH), 1,610 cm−1 (–N=CH). MS-FAB:(M + H)+1022. Anal.calcd.for C60H52N4O12: C, 70.58; H, 5.14; N, 5.49. Found: C, 70.39; H, 5.19; N, 5.45.
m-nitro benzaldehyde based resorcin[4]arenes Schiff base (4i): Yield 64%; M.P.160 °C; color: creamish. 1HNMR (CDCl3): δ = 1.63(d, 12H CH3), 4.64 (q, 4H bridge CH), 9.63(s, 8H –OH), 8.89(s, 4H –N=CH), 7.3–7.9(s, 20H ArH). IR (KBr): ν = 3,222 cm−1 (–OH), 1,606 cm−1 (–N=CH), 1508 (N–O). MS-FAB:(M + H)+1138. Anal.calcd.for C60H48N8O16: C, 63.38; H, 4.25; N, 9.85. Found: C, 62.29; H, 4.15; N, 9.70.
Results and discussion
In this article we report a new method for the derivatization of the resorcin[4]arenes at the upper rim by reaction with diazonium salts leading to the corresponding tetra azo compound in high yields which on reduction with Na2S2O4. gives amino groups at all the four positions of resorcin[4]arenes Tetra amino substituted resorcin[4]arene, which is very useful intermediate, suitable for the preparation of many other derivatives or aminoresorcin[4]arenes based receptors, should usually be synthesized in two steps (i) nitration of the upper rim and (ii) reduction of corresponding nitro compound. During our attempts for nitration of resorcin[4]arenes we found that the reaction conditions normally used in ‘classical’ chemistry did not work in this case. To afford the introduction of amino groups at the upper rim of resorcin[4]arenes we carried out a diazo coupling reaction between resorcin[4]arenes (1) and diazonium salt to yield the corresponding tetra azo compound. The reaction was carried out using different diazonium salts of aniline, p-nitro aniline and p-amino benzoic acid, However quantitative yields were obtained with the diazonium salt of p-amino benzoic acid because of ease in isolation. The addition of the diazonium salt to a solution of (1) led to an immediate colour change (yellow to red) indicating the formation of an azo compound. Stirring the reaction at room temperature gave colored precipitates which were recrystallised using alcohol. The structure of azo compound was confirmed by 1H NMR analysis, multiplet at δ 7.76–7.89 ppm due to the p-substituted phenyl rings, singlet at δ 9.87 (–OH & COOH group) and doublet at δ 1.65 ppm for CH3 and quartet at d 4.67 ppm for CH. The IR spectra gave C=O stretching at 1,698 cm−1 and hydrogen bonded –OH at 3,167 cm−1.
For reduction of the azo group we tried several different methods described in the literature [15, 16].The most efficient reduction of the azo compound (2) was achieved using Na2S2O4 and NaOH in an aqueous solution [17, 18]. Stirring the reaction mixture at elevated temp. (90 °C) gave the expected product as a pale yellow powder in 90% yield. The compound (3) was not stable in air for long therefore the reaction with different aromatic aldehydes (benzaldehyde, salisaldehyde, naphthaldehyde, 2-hydroxy naphthaldehyde, cinnamaldehyde, o-vanillin, vanillin, p-hydroxy benzaldehyde, m-nitro benzaldehyde) was carried out quickly in alcohol to yield corresponding Schiff-bases (4a–i), which were duly characterized. The synthetic reaction sequence is given in Scheme 1.
1H-NMR spectra of 4a–i revealed a singlet for azo methine (–N=CH) group at δ 8.8–8.9 ppm, a doublet at 1.65 ppm for (–CH3), a quartet at d 4.62–4.65 ppm for (–CH), a multiplet from 7.3 ppm to 7.9 ppm for aromatic protons (ArH), for (–OH) a singlet at δ 9.5–9.7 ppm. Compound 4f and 4g a singlet at δ 3.7 ppm for (–OCH3), Compound 4e a doublet at δ 4.6 ppm for (–CH=CH–). In the IR spectra, there is the typical absorption at 1,604–1,634 cm−1 for (–N=CH) and for (–OH) at 3,400–3,200 cm−1.
Conclusion
A new approach was developed to synthesize the tetra amino resorcin[4]arene. This amino derivative can serve as a key intermediate for further functionalization to obtain resorcin[4]arene with specific receptor properties. Nine new resorcin[4]arene Schiff-bases were synthesized and characterized. Binding properties of Schiff-bases of resorcin[4]arenes with various metal ions and different guests is being explored.
References
(a) Högberg, A.G.S.: Stereoselective synthesis and DNMR study of two 1,8,15,22-Tetraphenyl [14] metacyclophan-3,5,10,12,17,19,24,26-octols. J. Am. Chem. Soc. 102, 6046–6050 (1980); (b) Högberg, A.G.S.: Two steroisomeric macrocyclic resorcinol-acetaldehyde condensation products. J. Org. Chem. 45, 4498–4500 (1980); (c) Timmerman, P., Verboom, W., Reinhoudt, D.N.: Resorcinarenes. Tetrahedron. 52, 2663–2704 (1996)
Abis, L., Dalcanale, E., Du Vosel, A., Spera, S.: Structurally new macrocycles from the resorcinol-aldehyde condensation. Configurational and conformational analyses by means of dynamic NMR, NOE, and T1 experiments. J. Org. Chem. 53, 5475–5479 (1988)
(a) Cram, D.J., Cram, J.M.: Container molecules and their guests. Royal Society of Chemistry, Cambridge (1994); (b) Mandolini, L., Ungaro, R. (eds.): Calixarenes in Action. Imperial College Press, London (2000); (c) Atwood, J.L., Steed, J.W.: Encyclopedia of Supramolecular Chemistry. Marcel Dekker, Inc., New York (2004); (d) Jain, V.K., Mandalia, H.C.: The chemistry of calixpyrrole. Heterocycles 71, 1261–1314+2 (2007); (e) Jain, V.K., Pandya, R.A., Pillai, S.G., Agrawal, Y.K., Kanaiya, P.H.: Solid-phase extractive preconcentration and separation of lanthanum (III) and cerium (III) using a polymer-supported chelating Calix[4]arene Resin. J. Ana. Chem. 62, 104–112 (2007)
(a) Aoyama, Y., Tanaka, Y., Sugahara, S.: Molecular recognition. 5. Molecular recognition of sugars via hydrogen-bonding interaction with a synthetic polyhydroxy macrocycle. J. Am. Chem. Soc. 111, 5397–5404 (1989); (b) Aoyama, Y., Nonaka, Y., Tanaka, Y., Toi, H., Ogoshi, H.: Polar host–guest interactions. Solubilization of some polar compounds with lipophilic calix[6]arenes containing polar groups in a polar media. J. Chem. Soc., Perkin Trans. 2, 1025–1029, (1989); (c) Aoyama, Y., Tanaka, Y., Toi, H., Ogoshi, H. Polar Host-guest Interaction. Binding of nonionic polar compounds with a resorcinol- aldehyde cyclouligomer as a lipophilic polar host. J. Am. Chem. Soc. 110, 634–635 (1988); (d) Van-Velzen, E.U.T., Engbersen, J.F.J., Delagne, P.J., Mahy, J.W., Reinhoudt, D.N.: Self-assembled monolayers of resorcin[4]arene tetrasulfides on Gold. J. Am. Chem. Soc. 117, 6853–6862 (1995); (e) Laughrey, Z.R., Gibb, C.L.D., Senechal, T., Gibb, B.C.: Guest binding and orientation within open nanoscale hosts. Chem. Eur. J. 9, 130–139 (2003)
(a) Heinz, T., Rudkevich, D.M. Jr, Rebek, J.: Pairwise selection of guests in a cylindrical molecular capsule of nanometre dimensions. Nature. 394, 764–766 (1998); (b) Shivanyuk, A. Jr, Rebek, J.: Reversible encapsulation of multiple, neutral guests in hexameric resorcinarenehosts. Chem. Commun. 2424–2425, (2001); (c) Shivanyuk, A. Jr, Rebek, J.: Assembly of resorcinarene capsules in wet solvents. J. Am. Chem. Soc. 125, 3432–3433 (2003); (d) MacGillivray, L.R., Atwood, J.L.: A chiral spherical molecular assembly held together by 60 hydrogen bonds. Nature 389, 469–472 (1997); (e) Atwood, J.L., Barbour, L.J., Jerga, A.: Hydrogen-bonded molecular capsules are stable in polar media. Chem. Commun. 2376–2377 (2001); (f) Avram, L., Cohen, Y.: Hexameric capsules of lipophilic pyrogallolarene and resorcinarene in solutions as probed by diffusion NMR: one hydroxyl makes the difference. Org. Lett. 5(18), 3329–3332 (2003); (g) Avram, L., Cohen, Y.: Discrimination of guests encapsulation in large hexameric molecular capsules in solution: pyrogallol[4]arene versus resorcin[4]arene capsules. J. Am. Chem. Soc. 125, 16180–16181 (2003)
(a) Davis, F., Frary, E., Stirling, C.J.M.: Changing surface hydro- and oleophobicity with resorcinarene multilayers-a simple water/oil proofing process. Langmuir 20, 9075–9079 (2004) (b) Adams, H., Davis, F., Stirling, C.J.M.: Selective adsorption in gold–thiol monolayers of calix-4-resorcinarenes. J. Chem. Soc. Chem. Commun. 2527–2529 (1994); (c) Kurita, E., Fukushima, N., Fujimaka, M., Matsuzawa, Y., Kudo, K., Ichimura, K.: Macrocyclic amphiphiles. Part 2. Multi-point adsorptivity of the crown conformer of calix[4]resorcinarenes and their derivatives on surfaces of amorphous polar substrates. J. Mater. Chem. 8, 397–403 (1998)
Yamakawa, Y., Ueda, M., Nagahata, R., Takeuchi, K., Asai, M.: Rapid synthesis of dendrimers based on calix[4]resorcinarenes. J. Chem. Soc. Perkin Trans. 1, 4135–4139 (1998)
Antesberger, J., Cave, G.W.V., Ferrarelli, M.C., Heaven, M.W., Raston, C.L., Atwood, J.L.: Solvent-free, direct synthesis of supramolecular nano-capsules. Chem. Commun. 892–894 (2005)
Gualbert, J., Shahgaldian, P., Lazar, A., Coleman, A.W.: Solid lipid nanoparticles (SLNs): preparation and properties of Calix[4]resorcinarene-derived systems. J. Incl. Phenom. Macrocycl. Chem. 48, 37 (2004)
Yonetake, K., Nakayama, T., Ueda, M.: New liquid crystals based on calixarenes. J. Mater. Chem. 11, 761–767 (2001)
(a) Jain, V.K., Pillai, S.G., Kanaiya, P.H.: Octafunctionalized calix[4]resorcinarene-N-fenil-acetohydroxamic acid for the separation, preconcentration and transport studies of Cerium (IV). J. Braz. Chem. Soc. 17, 1316–1322 (2006); (b) Jain, V.K., Pillai, S.G., Pandya, R.A., Agrawal, Y.K., Shrivastav, P.S.: Selective extraction, separation and transport studies of thorium (IV) Using octa functionalized calix[4]resorcinarene-hydroxamic acid. Anal. Sci. 21, 129–135 (2005) (c) Jain, V.K., Pillai, S.G., Pandya, R.A., Agrawal, Y.K., Shrivastav, P.S.: Molecular octopus: octa functionalized calix[4]resorcinarene-hydroxamic acid [C4RAHA] for selective extraction, separation and preconcentration of U(VI). Talanta 2, 466–475 (2005)
Tunstad, L.M., Tucker, J.A., Dalcanale, E., Weiser, J., Bryant, J.A., Sherman, J.C., Helgeson, R.C., Knobler, C.B., Cram, D.J.: Host-guest complexation. 48. Octol building blocks for cavitands and carcerands. J. Org. Chem. 54, 1305–1312 (1989)
Dwyer, F.R., Mellor, D.P. (eds.): Chelating agents and metal chelates. Academic press, New York (1964)
Menon, S.K., Jogani, S.K., Agrawal, Y.K.: Macrocyclic schiff bases and their analytical application. Rev. Anal. Chem. 19, 361–412 (2000)
Ak, M.S., DeligÖz, H.: Azocalixarenes. 6: Synthesis, complexation, extraction and thermal behavior of four new azocalix[4]arenas. J. Incl. Phenom. Macrocycl. Chem. 59, 115–123 (2007)
Desroches, C., Parola, S., Vocanson, F., Perrin, M., Lamartine, R., Letoffe, J.M., Bouix, J.: Nitration of thiacalix[4]arene using nitrosium nitrate complexes: synthesis and characterization of tetranitro-, tetraamino-, and tetra(4-pyridylimino) tetrahydroxythiacalix[4]arene. New J. Chem. 26, 651–655 (2002)
Morita, Y., Agawa, T., Nomura, E., Taniguchi, H.: Synthesis and NMR behavior of calix[4]quinone and calix[4]hydroquinone. J. Org. Chem. 57, 3658–3662 (1992)
Lhotak, P., Moravek, J., Stibor, I.: Diazo coupling: an alternative method for the upper rim amination of thiacalix[4]arenas. Tetrahedron. Lett. 43, 3665–3668 (2002)
Acknowledgements
The authors gratefully acknowledge the financial assistance provided by Council of Scientific and Industrial Research (CSIR), New Delhi to carry out this work. The authors acknowledge CDRI (Lucknow), SAIF (IIT, Mumbai), IISC (Bangalore) for providing instrumental facilities and authors also acknowledge INFLIBNET, Ahmedabad for e-journals.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jain, V.K., Kanaiya, P.H. Diazo reductive: a new approach to the synthesis of novel “upper rim” functionalized resorcin[4]arene Schiff-bases. J Incl Phenom Macrocycl Chem 62, 111–115 (2008). https://doi.org/10.1007/s10847-008-9445-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10847-008-9445-1