A R T I C L E S
Tang et al.
arrays leads invariably to a single helical sense under any single
set of conditions. One could, in one way of thinking, see these
systems of biological origin as disadvantaged because they block
our observation of how helical sense is chosen. In many
synthetic systems, in contrast, helical sense control can be
addressed by using the power of chemistry, in its ability to
incorporate nonracemic chirality, therefore opening the potential
to observe the interplay of local chirality and helical sense.
In one synthetic polymer, this opportunity to observe how
helical sense is controlled leads to quantitative insight into the
cooperativity of helical systems.26,27 One aspect of this coop-
erativity is encountered in an experiment in which mirror-image-
related moieties compete for control of helical sense, giving
rise to a highly nonlinear relationship between the enantiomeric
excess of the competing groups and the excess helical sense.28,29
This experiment, termed “majority rule,” 28 is altered in the work
reported here to ask a new kind of question: What would happen
if the information offered to a helical array arose from a conflict
in which the chiral forces favoring left- and right-handed
conformations arose from structurally different enantiomers?
In answering this question, we discover, as outlined in the
work presented below, a new kind of relationship between
temperature and optical activity in both dilute solutions and
liquid crystals; this relationship can be described in quantitative
detail by statistical physics and could be applied, in principle,
to a wide variety of helical arrays.30
a polymer with a helical conformation that is interrupted infre-
quently by mobile helical reversal defects, allowing the chains
to dynamically interconvert between left- and right-handed con-
formations. The appended competing enantiomeric groups,
which may reside on all units of the chain or some fraction of
the units, then respectively favor the mirror-related helical
senses.
Lr
M ) erf
(p - p*)
(1)
[
]
x
2p*(1 - p*)
In the theoretical expression (eq 1),30 the excess of one helical
sense is expressed by M, which varies from unity for a single
helical sense, which would lead to the maximum chiral optical
effect, to zero for a 50:50 mixture, for which all chiral optical
measurements at all wavelengths would be zero. As shown in
eq 1, M is related through an error function (erf) to four different
variables. The term r measures the fraction of chiral groups
appended to the chain, allowing therefore for the possibility that
the competing chiral units are dispersed among achiral units. L
measures the number of helical units along the chain between
the interrupting helical reversal units and therefore is a measure
of the cooperativity.26,27 Finally there is a term measuring the
difference between the variables p and p*. If the competing
chiral units are Ca and Cb, then p measures the fraction of Ca,
that is, Ca/(Ca + Cb), which is determined by synthesis and
therefore can be varied.
The variable p* is defined by eq 2, where ∆Ga and -∆Gb
are the chiral bias energies by which the competing groups favor
one or the other helical sense. If Ca and Cb are enantiomers of
Theoretical Background
An approximate algebraic solution (eq 1) to a one-dimensional
Ising model, describing the effect of conflicting chiral informa-
tion on choosing helical sense in a helical array, has been devel-
oped to understand the experimental results for a helical polymer
appended with enantiomeric groups.28,31 The theory addresses
∆Ga
p* )
(2)
(
)
∆Ga + ∆Gb
each other, then these free energies must be identical in
magnitude, and therefore p* must equal 1/2. This situation gives
rise to the theoretical expressions originally derived for the
majority rule experiments28,31 and informs us that, for a 1:1
mixture of Ca and Cb, for which p ) 1/2, p - p* is zero and
therefore, from eq 1, M must be zero and so must all chiral
optical measures. This conclusion is independent of temperature,
although the free energy terms in eq 2 may be temperature-
dependent, because the temperature dependencies of ∆Ga and
∆Gb must be identical. In other words, p* is independent of
temperature when the competing chiral groups are mirror-
related. For the latter situation, the theory has been shown to
quantitatively fit experiment.31
(19) Nomura, R.; Tabei, J.; Masuda, T. J. Am. Chem. Soc. 2001, 123, 8430.
(20) Li, C. Y.; Cheng, S. Z. D.; Weng, X.; Ge, J. J.; Bai, F.; Zhang, J. Z.;
Calhoun, B. H.; Harris, F. W.; Chien, L.-C.; Lotz, B. J. Am. Chem. Soc.
2001, 123, 2462.
(21) Tanatani, A.; Mio, M. J.; Moore, J. S. J. Am. Chem. Soc. 2001, 123, 1792.
See also: Zhao, D.; Moore, J. S., Macromolecules 2003, 36, 2712.
(22) (a) Scho¨ning, K.-U.; Scholz, P.; Guntha, S.; Wu, X.; Krishnamurthy, R.;
Eschenmoser, A. Science 2000, 290, 1347 (also see Orgel, L., on p 1306
for comments). (b) Kozlov, I. A.; Orgel, L. E.; Nielsen, P. E. Angew. Chem.,
Int. Ed. 2000, 39, 4292.
(23) Kramer, R.; Lehn, J.-M.; Marquis-Rigault, A. Proc. Natl. Acad. Sci. U.S.A.
1993, 90, 5394.
(24) For several review articles on synthetic helical arrays, see: Green, M. M.;
Meijer, E. W.; Nolte, R. L. M., Materials-Chirality. Top. Stereochem. 2003,
in press.
(25) (a) Ismagilov, R. F.; Schwartz, A.; Bowden, N.; Whitesides, G. M. Angew.
Chem., Int. Ed. 2002, 41, 652. (b) Beier, M.; Reck, F.; Wagner, T.;
Krishnamurthy, R.; Eschenmoser, A. Science 1999, 283, 699. (c) Inai, Y.;
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2002, 124, 2466. (d) Declercq, R.; Aerschot, A. V.; Read, R. J.; Herdewijin,
P.; Meervelt, L. V. J. Am. Chem. Soc. 2002, 124, 928. (e) Woll, M. G.;
Fisk, J. D.; LePlae, P. R.; Gellman, S. H. J. Am. Chem. Soc. 2002, 124,
12447. (f) Pu, L. Chem. ReV. 1998, 98, 2405.
(26) Green, M. M.; Peterson, N. C.; Sato, T.; Teramoto, A.; Lifson, S. Science
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S.; Selinger, R. L. B.; Selinger, J. V. Angew. Chem., Int. Ed. 1999, 38,
3138.
In the current work, we are interested in the situation where
Ca and Cb are structurally different enantiomers,30 that is, not
enantiomers of each other. It follows that ∆Ga and ∆Gb might
have different temperature dependencies, and therefore p* would
change with temperature. This sets up an interesting experiment.
If Ca and Cb are randomly distributed along the chain in a
proportion corresponding to p, then there may be a temperature,
Tc, at which p* will be equal to p and therefore a temperature
at which M ) 0. However, above and below Tc, p and p* will
differ because p* is a function of temperature while p is not,
causing M to differ from zero.
(27) Green, M. M. In Circular DichroismsPrinciples and Application, 2nd ed.;
Berova, N., Nakanishi, K., Woody, R. W., Eds.; Wiley-VCH: New York,
2000; Chapter 17.
(28) (a) Green, M. M.; Garetz, B. A.; Munoz, B.; Chang, H.; Hoke, S.; Cooks,
R. G. J. Am. Chem. Soc. 1995, 117, 4181. (b) Li, J.; Schuster, G. B.; Cheon,
K.-S.; Green, M. M.; Selinger, J. V. J. Am. Chem. Soc. 2000, 122, 2603.
(29) Downie, A. R.; Elliott, A.; Hauby, W. E.; Malcom, B. R. Proc. R. Soc.
London A 1957, 242, 325. Morawetz, H. Macromolecules in Solution, 2nd
ed.; Wiley-Interscience: New York, 1975; S. 250-251. See also: Heitz,
F.; Spach, G. Macromolecules 1971, 4, 429; Macromolecules 1975, 8, 740.
(30) For a preliminary report of this work, see: Cheon, K. S.; Selinger, J. V.;
Green, M. M. Angew. Chem., Int. Ed. 2000, 39, 1482.
The sign of M will reflect the sign of the term p - p*, and
this sign will change above and below Tc. Therefore, the sense
of the helix in excess will, as well, change above and below Tc.
(31) Selinger, J. V.; Selinger, R. L. B. Phys. ReV. Lett. 1996, 76, 58.
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7314 J. AM. CHEM. SOC. VOL. 125, NO. 24, 2003