The Cp2ZrCl2-catalyzed cycloalumination of functionally substituted olefins with triethylaluminum

Article abstract of DOI:10.1007/s11172-011-0243-3

Reactions of functionally substituted olefins (allylamines, sulfides and ethers, homoallylic alcohols and amines, as well as vinyl ethers) with Et ₃Al in the presence of Cp₂ZrCl₂ as a catalyst were studied. Cycloalumination of allylamines occurs with high regioselectivity to furnish after subsequent deuterolysis 4-deutero-2-(deuteromethyl)butyl- substituted amines. Cycloalumination of alkyl allyl sulfide is accompanied by a side process of the C-S bond cleavage. In the case of allyl and vinyl ethers, no cycloalumination products are formed under the reaction conditions. However, the reactions with homoallylic alcohol and amine after deuterolysis gave the corresponding dideutero-containing compounds in good yields.

Full text of DOI:10.1007/s11172-011-0243-3

Russian Chemical Bulletin, International Edition, Vol. 60, No. 8, pp. 1628—1632, August, 2011
1628
The Cp2ZrCl2ꢀcatalyzed cycloalumination
of functionally substituted olefins with triethylaluminum
I. R. Ramazanov, R. N. Kadikova, and U. M. Dzhemilev
Institute of Petrochemistry and Catalysis, Russian Academy of Sciences,
141 prosp. Oktyabrya, 450075 Ufa, Russian Federation.
Fax: +7 (347) 284 2750. Eꢀmail: iramazan@inbox.ru
Reactions of functionally substituted olefins (allylamines, sulfides and ethers, homoallylic
alcohols and amines, as well as vinyl ethers) with Et3Al in the presence of Cp2ZrCl2 as a catalyst
were studied. Cycloalumination of allylamines occurs with high regioselectivity to furnish after
subsequent deuterolysis 4ꢀdeuteroꢀ2ꢀ(deuteromethyl)butylꢀsubstituted amines. Cycloaluminaꢀ
tion of alkyl allyl sulfide is accompanied by a side process of the C—S bond cleavage. In the case
of allyl and vinyl ethers, no cycloalumination products are formed under the reaction condiꢀ
tions. However, the reactions with homoallylic alcohol and amine after deuterolysis gave the
corresponding dideuteroꢀcontaining compounds in good yields.
Key words: cycloalumination, functionally substituted olefins, aluminacyclopentanes.
Reactions of αꢀolefins and acetylenes with Et3Al cataꢀ
lyzed with Cp2ZrCl2 and known as a cycloalumination
reaction is a unique method for the synthesis of aluminaꢀ
cyclopentanes and aluminacyclopentenes.1,2 Zirconium
chiral complexes Cp(NMI)ZrCl2 (NMI is the 1ꢀneoꢀ
menthylindenyl) and [ETBHI]ZrC12 (ETBHI is the
(R,R)ꢀethyleneꢀ1,2ꢀbis(η5ꢀ4,5,6,7ꢀtetrahydroꢀ1ꢀindenyl))
catalyze reactions of Et3Al with allyl phenyl sulfide and
2,5ꢀdihydrofuran.3 However, there are still absent the data
concerning cycloalumination of vinyl, allyl, and homoꢀ
allylic alcohols and amines, whose transformation would
have allowed one to develop new oneꢀpot methods for the
synthesis of functionally substituted cyclobutanes, cycloꢀ
pentanols, and tetrahydrothiophenes.4 Earlier,5 we have
studied a Cp2ZrCl2ꢀcatalyzed reaction of acetylene alcoꢀ
hols and propargylamines with Et3Al. It was found that
the presence of a hydroxy or amine function in the
molecule of acetylene compound does not interfere
cycloalumination and formation of 2,3ꢀdisubstituted
aluminacyclopentꢀ2ꢀenes. In this connection and in order
to develop an efficient method for the synthesis of
functionally substituted aluminacyclopentanes, we studied
the reaction of vinyl, allyl, and homoallylic compounds
with Et3Al in the presence of Cp2ZrCl2 in catalytic
amounts.
bond proceeds regioselectively. The structures of the
deuterolysis products were established by oneꢀ and twoꢀ
dimensional NMR spectroscopy. The 13C NMR spectrum
of compound 1a exhibits two triplets for the deuteroꢀsubꢀ
stituted carbon atoms at δ 11.00 (1JC,D = 19 Hz) and
17.45 (1JC,D = 19 Hz). Two multiplets for the diaꢀ
stereotopic hydrogen atoms of the methylene group at the
nitrogen atom have a crossꢀpeak with the hydrogen atom
on the tertiary carbon atom at the CH2D group in the
COSY spectrum. In the HMBC spectrum, the multiplets
mentioned have crossꢀpeaks with the signal for the tertiary
carbon atom of the cyclohexane ring, the methylene carꢀ
bon atom of the 2ꢀdeuteroethyl fragment, the tertiary carꢀ
bon atom at the CH2D group, as well as with the triplet
signal for the CH2D group at δ 17.45. Compounds 1b,c
were identified similarly. The numeration of the carbon
atoms used in the description of the NMR spectra of
deuterolysis products 1a—c is shown in Fig. 1.
It is interesting to note that allylphenylamine and
diallylphenylamine showed no activity in the reaction unꢀ
der study. Unsubstituted diallylamine gave cycloaluminaꢀ
tion product straight at the two double bonds. After deuꢀ
terolysis of the reaction mixture, the tetradeuteroꢀsubstiꢀ
tuted amine 2 was isolated in 69% yield. However, the
reaction with diallyl(tertꢀoctyl)amine involves only one
double bond to yield after deuterolysis a dideuteroꢀconꢀ
taining unsaturated amine 3 (see Scheme 1).
It was found that the reaction of allylamines (Nꢀallylꢀ
cyclohexylamine, Nꢀallylꢀtertꢀoctylamine, Nꢀallylpiperꢀ
idine) with 1 equiv. of Et3Al in the presence of Cp2ZrCl2
(10 mol.%) in hexane for 8 h at 40 °C after subsequent
deuterolysis leads to the corresponding 4ꢀdeuteroꢀ2ꢀ
(deuteromethyl)butylꢀsubstituted amines 1a—c in 64—83%
yields (Scheme 1). Cycloalumination of the unsaturated
Catalytic cycloalumination of allyl heptyl sulfide gives
lower yields (Scheme 2). After deuterolysis of the reaction
mixture, d1ꢀheptanethiol 5 was found in the mixture (the
yield was 28%) together with dideuterated derivative 4
(the yield was 53%). The 13C NMR spectrum of comꢀ
Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 8, pp. 1602—1606, August, 2011.
1066ꢀ5285/11/6008ꢀ1628 © 2011 Springer Science+Business Media, Inc.

Zrꢀcatalyzed cycloalumination of alkenes
Russ.Chem.Bull., Int.Ed., Vol. 60, No. 8, August, 2011
1629
Scheme 1
pound 4 exhibits two triplets for the CH2D group at δ 11.01
(1JC,D = 19 Hz) and 18.66 (1JC,D = 19 Hz).
group to elimination, we failed to involve alkyl allyl and
alkyl vinyl ethers into the cycloalumination reaction. In
the reaction of allyl nꢀundecyl ether with Et3Al in the
presence of a catalytic amount of Cp2ZrCl2 under condiꢀ
tions indicated above after subsequent hydrolysis of the
reaction mixture, undecyl alcohol was obtained in quantiꢀ
tative yield, whereas butyl vinyl ether in the reaction under
study after hydrolysis gave nꢀbutyl alcohol.
However, when homoallylic alcohol (3ꢀbutenꢀ1ꢀol)
was involved into the reaction with subsequent deuteroꢀ
lysis of the reaction mixture, a deuteroꢀcontaining alcohol 6
was obtained in 79% yield (see Scheme 2). Thus, when two
methylene units are present between a double bond and
a hydroxy group, cycloalumination is not accompanied by
a side elimination reaction. Substituted homoallylic amine
(1ꢀ(3ꢀbutenyl)piperidine) behaves similarly in the cycloꢀ
alumination reaction. After deuterolysis, a dideuteroꢀconꢀ
taining amine 7 was isolated in good yield (85%).
According to the mechanism of the cycloalumination
reaction suggested earlier,1,6 one of the steps consists in the
formation of zirconoceneꢀethylene complex A (Scheme 3),
which by the reaction with allyl sulfide is converted either
to zirconacyclopropane B, or to zirconacyclopentane C.
Apparently, heptanethiol is formed by βꢀelimination of
a sulfide group from the intermediate B. Such a phenomeꢀ
non has been observed earlier7 in the reaction of Negishi
reagent with allyl ethers, therefore, we assume that cleavꢀ
age of the carbon—oxygen bond will occur in the case of
allyl and vinyl ethers under condition of the studied by us
reaction. In fact, because of higher disposition of an alkoxy
Experimental
Commercially available reactants were used in this work.
Reactions with organoaluminum compounds were carried out
under dry argon. Hexane was distilled over Bui3Al. Monoꢀ and
diallylamines were obtained by the reaction of allyl bromide
with secondary and primary amines upon the action of NaH.8
Allyl heptyl sulfide was synthesized by allylation of heptaneꢀ
thiol.9 1ꢀ(3ꢀButenyl)piperidine was obtained by aminomethylaꢀ
tion of allyl bromide.10 Reaction products were analyzed on
a Carlo Erba chromatograph (using a 25 m × 0.2 mm Ultraꢀ1 glass
Fig. 1. The numeration of carbon atoms used in the description
of the NMR spectra of the deuterolysis products 1a—c.

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Ramazanov et al.
Scheme 2
Scheme 3

Zrꢀcatalyzed cycloalumination of alkenes
Russ.Chem.Bull., Int.Ed., Vol. 60, No. 8, August, 2011
1631
capillary column (Hewlett Packard), flameꢀionizing detector,
operating temperature of 50—170 °C, and helium as a carrier
gas). Mass spectra were measured on a Finnigan 4021 instruꢀ
ment (70 eV), the temperature of injector was 200 °C. Elemental
analysis was performed on a Carlo Erba model 1106 elemental
analyser. 1H and 13C NMR spectra were recorded on a Bruker
Avance 400 spectrometer (400.13 and 100.62 MHz, respectively),
using SiMe4 and CDCl3, respectively, as internal standards. The
product yields were determined by GLC using undecane or hexaꢀ
decane as internal standards.
Cycloalumination of Nꢀallylamines, N,Nꢀdiallylamines, and allyl
heptyl sulfide (general procedure). Allylic compound (1 mmol),
Cp2ZrCl2 (0.029 g, 0.1 mmol), hexane (3 mL), and Et3Al
(1 mmol, or 2 mmol in the case of N,Nꢀdiallylamines) were
sequentially placed under argon into a 50ꢀmL glass reactor
equipped with a magnetic stirrer and heated using a water bath
to 40 °C, and the mixture was stirred for 8 h at 40 °C.
Bis[4ꢀdeuteroꢀ2ꢀ(deuteromethyl)butyl]amine (2). Clear oily
liquid. The yield was 69%, b.p. 78—80 °C (15 Torr). Found (%):
C, 73.93; N, 8.51. C10H19D4N. Calculated (%): C, 74.45;
N, 8.68. 1H NMR, δ: 0.85—0.95 (m, 8 H, C(1)H2D, C(4)H2D);
1.10—1.20 (m, 2 H, C(3)Ha); 1.35—1.50 (m, 2 H, C(3)Hb);
1.50—1.60 (m, 2 H, C(2)H); 2.30—2.45 (m, 2 H, C(5)Ha);
2.45—2.60 (m, 2 H, C(5)Hb). 13C NMR, δ: 11.02 (t, 2 C(4),
1JC,D = 19 Hz); 17.34 (t, 2 C(1), 1JC,D = 19 Hz); 27.49 (2 C(3));
34.75 (2 C(2)); 56.32 (2 C(5)).
NꢀAllylꢀNꢀ[4ꢀdeuteroꢀ2ꢀ(deuteromethyl)butyl]ꢀtertꢀoctylꢀ
amine (3). Clear oily liquid. The yield was 80%, b.p. 101—103 °C
(1 Torr). Found (%): C, 79.14; N, 5.71. C16H31D2N. Calculatꢀ
ed (%): C, 79.59; N, 5.80. 1H NMR, δ: 0.80—0.90 (m, 4 H,
C(1)H2D, C(4)H2D); 1.01 (s, 9 H, C(12)H3, C(15)H3, C(16)H3);
1.15 (s, 6 H, C(13)H3, C(14)H3); 1.10—1.20 (m, 1 H, C(3)Ha);
1.44 (s, 2 H, C(10)H2); 1.35—1.60 (m, 2 H, C(2)H, C(3)Hb);
2.20—2.30 (m, 1 H, C(5)Ha); 2.30—2.40 (m, 1 H, C(5)Hb);
4.90—5.00 (m, 1 H, C(8)Ha); 5.05—5.15 (m, 1 H, C(8)Hb);
5.85—6.00 (m, 1 H, C(7)H). 13C NMR, δ: 11.04 (t, C(4),
Deuterolysis of aluminacyclopentanes (general procedure).
Hexane (5 mL) was added to the reaction mixture obtained by
the procedure described above, followed by a dropwise addition
of D2O (3 mL) with cooling of the reactor in an ice bath.
A precipitate that formed was filtered off using a paper filter.
The aqueous layer was extracted with diethyl ether, the extract
was combined with the organic layer, dried with anhydrous
CaCl2, and concentrated in vacuo. Individual compounds 1a—c
and 2—5 were isolated by distillation at reduced pressure.
1ꢀ[4ꢀDeuteroꢀ2ꢀ(deuteromethyl)butyl]cyclohexylamine (1a).
Clear oily liquid. The yield was 64%, b.p. 93—95 °C (10 Torr).
Found (%): C, 75.93; N, 8.01. C11H21D2N. Calculated (%):
1
1JC,D = 19 Hz); 17.12 (t, C(1), JC,D = 19 Hz); 27.42 (C(3));
28.05, 28.18 (C(13), C(14)); 34.59 (C(2)); 49.52 (C(5)); 53.96
(C(10)); 56.98 (C(6)); 113.50 (C(8)); 141.21 (C7)). MS, m/z:
241 [M]+ (1), 226 [M – CH3]+ (9), 182 [C8H17tN(All)CH2]+
(6), 171 (12), 170 (92), 169 (12).
4ꢀDeuteroꢀ2ꢀ(deuteromethyl)butyl heptyl sulfide (4). Clear
oily liquid. The yield was 53%, b.p. 97—101 °C (5 Torr). Found (%):
C, 69.11. C12H24D2S. Calculated (%): C, 70.51. 1H NMR, δ:
0.80—0.90 (m, 5 H, C(4)H2D, C(12)H3); 0.90—1.00 (m, 2 H,
C(1)H2D); 1.15—1.40 (m, 9 H, C(3)Ha, C(8)H2—C(11)H2);
1.40—1.50 (m, 1 H, C(3)Hb); 1.50—1.65 (m, 3 H, C(2)H,
C(7)H2); 2.30—2.40 (m, 1 H, C(5)Ha); 2.48 (t, 2 H, C(6)H2,
J = 7.6 Hz); 2.45—2.55 (m, 1 H, C(5)Hb). 13C NMR, δ: 11.00
1
C, 77.12; N, 8.18. H NMR, δ: 0.85—0.95 (m, 4 H, C(1)H2D,
C(4)H2D); 1.00—1.35 (m, 6 H, C(3)Ha, C(7)Ha—C(11)Ha);
1.35—1.45 (m, 1 H, C(3)Hb); 1.45—1.55 (m, 1 H, C(2)H);
1.60—1.65 (m, 1 H, C(9)Hb); 1.70—1.80 (m, 2 H, C(8)Hb,
C(10)Hb); 1.85—1.95 (m, 2 H, C(7)Hb, C(11)Hb); 2.35—2.45
(m, 2 H, C(5)Ha, C(6)H); 2.50—2.60 (m, 1 H, C(5)Hb). 13C NMR,
δ: 11.00 (t, C(1), 1JC,D = 19 Hz); 17.45 (t, C(4), 1JC,D = 19 Hz);
25.14 (2 C, C(8), C(10)); 26.24 (C(9)); 27.49 (C(3)); 33.67, 33.76
(2 C, C(7), C(11)); 34.93 (C(2)); 53.19 (C(5)); 56.99 (C(6)).
MS, m/z: 172 [M + 1]+ (2), 171 [M]+ (6), 170 [M – 1]+ (2),
141 [M – Et]+ (1), 128 (18), 127 (6), 113 (10), 112 (100)
[cycloꢀC6H11NHCH2]+.
1
1
(t, C(4), JC,D = 19 Hz); 14.04 (C(12)); 18.20 (t, C(1), JC,D
=
= 19 Hz); 22.60 (C(11)); 28.69 (C(3)); 28.90, 28.92 (C(8), C(9));
29.79 (C(7)); 31.74 (C(10)); 32.84 (C(6)); 34.84 (C(2)); 39.83 (C(5)).
d1ꢀHeptanethiol (5). Clear oily liquid. The yield was 28%,
1
b.p. 41—45 °C (5 Torr). H and 13C NMR spectral characterisꢀ
tics are identical to those described earlier11 (with allowance for
the absence of signals for SH group in the 1H NMR spectrum).
Cycloalumination of 3ꢀbutenꢀ1ꢀol and 1ꢀ(3ꢀbutenyl)piperidine
(general procedure). A homoallylic compound (1 mmol), Cp2ZrCl2
(0.015 g, 0.05 mmol), hexane (3 mL), and Et3Al (2 mmol) were
sequentially placed under argon into a 50ꢀmL glass reactor
equipped with a magnetic stirrer and heated using a water bath
to 40 °C, and the mixture was stirred for 6 h at 40 °C. Deuteroꢀconꢀ
taining compounds were obtained according to the procedure
described above for the deuterolysis of aluminacyclopentanes.
5ꢀDeuteroꢀ3ꢀ(deuteromethyl)ꢀ1ꢀdeuteroxypentane (6). Clear
oily liquid. The yield was 79%, b.p. 63—65 °C (20 Torr). Found
1ꢀ[4ꢀDeuteroꢀ2ꢀ(deuteromethyl)butyl]ꢀtertꢀoctylamine (1b).
Clear oily liquid. The yield was 79%, b.p. 59—61 °C (1 Torr).
Found (%): C, 76.21; N, 6.17. C13H27D2N. Calculated (%):
1
C, 77.53; N, 6.96. H NMR, δ: 0.85—0.95 (m, 4 H, C(1)H2D,
C(4)H2D); 1.02 (s, 9 H, C(9)H3, C(12)H3, C(13)H3); 1.13 (s, 6 H,
C(10)H3, C(11)H3); 1.10—1.20 (m, 1 H, C(3)Ha); 1.43 (s, 2 H,
C(7)H2); 1.40—1.50 (m, 2 H, C(2)H, C(3)Hb); 2.25—2.60 (m, 2 H,
1
C(5)H2). 13C NMR, δ: 11.06 (t, C(4), JC,D = 19 Hz); 17.62
1
(C(1)); 27.68 (C(3)); 28.84, 28.95 (2 C, C(10), C(11)); 31.80
(3 C, C(9), C(12), C(13)); 35.84 (C(2)); 48.01 (C(5)); 53.59 (C(7));
53.94 (C(6)).
(%): C, 67.82. C6H11D3O. Calculated (%): C, 68.51. H NMR,
δ: 0.60—1.00 (m, 4 H, C(1)H2D, C(4)H2D); 1.00—1.80 (m, 5 H,
C(3)H2, C(5)H2, C(2)H); 3.63 (t, 2 H, C(6)H2, J = 5.6 Hz).
13C NMR, δ: 10.84 (t, C(4), 1JC,D = 19 Hz); 18.72 (t, C(1), 1JC,D
=
1ꢀ[4ꢀDeuteroꢀ2ꢀ(deuteromethyl)butyl]piperidine (1c). Clear
oily liquid. The yield was 83%, b.p. 65—67 °C (10 Torr). Found (%):
C, 76.04; N, 8.99. C10H19D2N. Calculated (%): C, 76.36; N, 8.90.
1H NMR, δ: 0.80—0.90 (m, 4 H, C(1)H2D, C(4)H2D); 1.00—1.15
(m, 2 H, C(3)H2); 1.35—1.50 (m, 3 H, C(2)H, C(8)H2); 1.50—1.65
(m, 4 H, C(7)H2, C(8)H2); 1.95—2.15 (m, 2 H, C(5)H2); 2.20—2.40
= 19 Hz); 29.40 (C(3)); 30.96 (C(2)); 39.43 (C(5)); 60.98 (C(6)).
1ꢀ[5ꢀDeuteroꢀ3ꢀ(deuteromethyl)pentyl]piperidine (7). Clear
oily liquid. The yield was 85%, b.p. 70—74 °C (5 Torr). Found (%):
C, 77.51; N, 8.08. C11H21D2N. Calculated (%): C, 77.12; N, 8.18.
1H NMR, δ: 0.80—0.95 (m, 4 H, C(1)H2D, C(4)H2D); 1.10—1.50
(m, 7 H, C(2)H, C(3)H2, C(5)H2, C(9)H2); 1.55—1.65 (m, 4 H,
C(8)H2, C(10)H2); 2.25—2.35 (m, 2 H, C(6)H2); 2.35—2.45
(m, 4 H, C(6)H2, C(10)H2). 13C NMR, δ: 11.11 (t, C(4), 1JC,D
= 19 Hz); 17.66 (t, C(1), JC,D = 19 Hz); 24.65 (C(8)); 26.08
=
1
(2 C, C(7), C(9)); 27.82 (C(3)); 31.88 (C(2)); 66.38 (C(5)).
(m, 4 H, C(7)H2, C(11)H2). 13C NMR, δ: 11.01 (t, C(4), 1JC,D
=

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Russ.Chem.Bull., Int.Ed., Vol. 60, No. 8, August, 2011
Ramazanov et al.
1
= 19 Hz); 19.05 (t, C(1), JC,D = 19 Hz); 24.55 (C(9)); 26.03
(2 C, C(8), C(10)); 29.56 (C(3)); 33.21, 33.60 (C(5), C(6)); 54.78
(2 C, C(7), C(11)); 57.81 (C(6)).
Khim., 1989, 207 [Bull. Acad. Sci. USSR, Div. Chem. Sci.
(Engl. Transl.), 1989, 38, 194].
2. U. M. Dzhemilev, A. G. Ibragimov, A. P. Zolotarev, Menꢀ
deleev Commun., 1992, 135.
Reaction of allyl nꢀundecyl and butyl vinyl ethers with Et3Al
catalyzed with Zr (general procedure). nꢀHexadecane (0.5 mmol),
allyl nꢀundecyl or butyl vinyl ethers (1 mmol), Cp2ZrCl2 (0.029 g,
0.1 mmol), hexane (3 mL), and Et3Al (1 mmol) were sequentialꢀ
ly placed under argon into a 50ꢀmL glass reactor equipped with
a magnetic stirrer and heated using a water bath to 40 °C, and the
mixture was stirred for 8 h at 40 °C. Hexane (5 mL) was added to
the reaction mixture, followed by a dropwise addition of H2O
(3 mL) cooling the reactor in an ice bath. A precipitate formed
was filtered off on a paper filter. The aqueous layer was extracted
with diethyl ether, the extract was combined with the organic
layer, dried with anhydrous CaCl2, and concentrated in vacuo.
Individual compounds (nꢀbutanol in the case of butyl vinyl ether and
nꢀundecanol in the case of allyl nꢀundecyl ether) were isolated
by distillation and identified by comparison of their 1H and 13C NMR
spectra with the reported data.11 The reaction product yields were
determined by GLC using nꢀhexadecane as an internal standard.
3. G. Dawson, C. A. Durrant, G. G. Kirk, R. J. Whitby, R. V. H.
Jones, M. C. H. Standen, Tetrahedron Lett., 1997, 38, 2335.
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134 [Russ. Chem. Rev. (Engl. Transl.), 2000, 69, 121].
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Akad. Nauk, Ser. Khim., 2011, 96 [Russ. Chem. Bull., Int.
Ed., 2011, 60, 99].
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T. Takahashi, J. Am. Chem. Soc., 1996, 118, 9577.
7. C. J. Rousset, D. R. Swanson, F. Lamaty, E. Negishi, Tetraꢀ
hedron Lett., 1989, 30, 5105.
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68, 646.
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11. SDBSWeb: http://riodb01.ibase.aist.go.jp/sdbs/ (National
Institute of Advanced Industrial Science and Technology,
13.12.2010).
This work was financially supported by the Ministry of
Education and Science of the Russian Federation (Grant
NShꢀ4105.2010.3).
References
1. U. M. Dzhemilev, A. G. Ibragimov, A. P. Zolotarev, R. R.
Muslukhov, G. A. Tolstikov, Izv. Akad. Nauk SSSR, Ser.
Received January 21, 2011;
in revised form April 7, 2011