Langmuir 1991, 7, 567-568
567
Elastic Theory of Helical Fibers W. Helfrich Fachbereich Ptcysik, Freie Universitat, Arnimallee 14, 0-1000 Berlin 33, Germany Received June 6, 1990 It is shown theoretically that fibers of isotropic cross section may be helical in their stable state if they consist of chiral molecules. Strips of crystalline bilayer in water can be helically wound (as if wrapped around an imaginary cylinder) or twisted about their central axis if they consist of chiral amphiphile~.l-~The forces giving rise to these helical structures are known: Anisotropic bilayers composed of chiral molecules, having no symmetry but a 2-fold axis of rotation in their plane, tend to curvelike a saddle or, where this is not possible, like a cylinder. Moreover, the edge of a bilayer of chiral molecules rnay possess a spontaneous torsion promoting the formation of helical and twisted ribbons. These "chiral" forces are balanced, in the stable state, by the forces due to the deformations of the strip and of the edges which are proportional to curvature and torsion, respectively. A theory of helical ribbons based on these concepts has been given el~ewhere.~ Chiral amphiphiles precipitating from aqueous solution also produce fibers of roundish cross section.'g2 Usually, the fibers are coiled to form narrow helices or staircases and their diameters of at least 10 nm are much too large for a single cylindrical micelle. Very thin fibers displaying a weak helical modulation were found with N-octyl-D-gluconamide by Fuhrhop et at.5 'Their width looks uniform in the electron micrographs, depending little or not on the phase of the spiral, but they may still be twice as thick as the 3.5 nm expected for a single cylindrical micelle. The existence of helical fibers that seem too round to be ribbons raises the general question whether or not an originally straight micelle of isotropic and, thus, circular cross section can prefer a helical state if it is made of chiral molecules. In the following, we will develop a theory of elasticity for such fibers which answers the question in the affirmative. It will be necessary to include energy terms of up to fourth order in curvature and torsion. The elastic energies of curvaiture and torsion have been considered for polymers by Bugl and Fujita.'j In addition to terms quadratic in the two st rains, their ansatz contains linear ones, which implies spontaneous curvature and spontaneous torsion. There is no term coupling the two deformations, but torsion automatically alters the plane of curvature along the polymer. In our model, all planes of curvature of the isotropically cylindrical micelle are equivalent and, as a consequence, there can be no spontaneous curvature of any kind. We will be interested only in external torsion which is that of the mathematical curve represented by the center line of the micelle. This torsion does not exist without. curvature to which it can be energetically coupled. Any internal torsion, i.e. twist
about the axis of the fiber, is assumed to be always at its equilibrium value which may depend on curvature. We view the micelle as a mathematical line, r = r(s), where s is the arc length. The strains may be expressed by the vector derivatives dnt/dsn (n = 1,2,...) of the tangent unit vector t(s) = dr/ds to the line. From the first four derivatives we construct independent scalars of up to fourth order in d/ds. For reasons of symmetry, only scalars invariant under simultaneous reversal of s and t can occur in the elastic energy associated with the strains. Evidently, there is no first-order invariant. Of the two qualdratic invariants, t-d2t/ds2and (dt/ds)2,only one is independent as follows from partial integration and
(1) Nakishima, N.; Asakuma, S.; Kunitake,T. J.Am. Chem. SOC.1985, 107, 509, and references cited therein. (2) Pfannemuller, B.; Welte, W. Chem. Phys. Lipids 1985, 37, 227. (3) Fuhrhop, J.-H.; Schnieder, P.; I3oekema, E.; Helfrich, W. J. Am. Chem. SOC.1988, 220, 2861. (4) Helfrich, W.; Prost, J. Phys. Rtw. A 1988, 38, 3065. (5)Fuhrhop, J.-H.; Schnieder, P.; Flosenberg, J.; Boekema, E. J. Am. Chem. SOC.1987, 109, 3387. (6) Bugl, P.; Fujita, S. J. Chem. Phys. 1969, 50, 3137.
d (dt -.- d2t)
0743-7463/91/2407-0567$02.50/0
t*dt/ds = 0 For symmetry reasons we choose
(1)
dt dt -.ds ds The only third-order invariant is
t.(
x
9)
(3)
The occurrence of a vector product indicates that this term changes sign together with molecular chirality,, Of the many fourth-order invariants, just a few are independent. We select (4)
and
(t x 9 ) '
(5)
The obvious relationship d2t d2t ds2 ds2 and (1)make the square of the second derivative depend on (4) and ( 5 ) . The invariant t.d4t/ds4 can be converted by partial integration into
"(ds t. ")ds3 and dt/ds.d3t/ds. Similarly, the latter invariant splits into (7) ds ds ds and d2t/ds2.d2t/ds2. This means that a complete elastic theory contains the derivatives ( 6 ) and (7). Fourth-order invariants containing two subsequent vector products can be converted into expressions with scalar products only. 0 1991 American Chemical Society
568 Langmuir, Vol. 7, No. 3, 1991
Helfrich
It is also possible to show that
x
Collecting the invariants (2) to (5) and multiplying them by elastic moduli k and (arbitrary) numerical factors, we obtain for the total fourth-order elastic energy per unit length (of micelle
y=-1 k - .dt - + kd3t t . 2 :!ds ds
(dt -X- d2t)+$ ds ds2
22
(-.-) d t dt ds ds
2+
The derivatives (6) and (7) have now been omitted. In an integral of y over s from s1 to s2, they yield boundary terms such as
which are negligible for long helical micelles comprising many periods. Using the curvature x = (dt/ds(and the torsion = t*(dt/ds X d2t/ds2)/X2 as defined in differential geometry, we can write eq 8 in the form 7
The final result may also be obtained directly from detailed symmetry considerations. Minimizing y with respect to 7 at fixed x leads, with dxlds = 0, to = -k3/k4
(10)
Inserting this into eq 9 gives
Evidently, a helix with the torsion over the straight state whenever
TO
k,' > k2k4 Minimizing eq 11 with respect to equilibrium curvature of the helix Xeq2 =
k32/k4
will be preferred
x
- k2
k22
(12) yields for the
(13)
For a helix in the usual parametric representation
r = ( a cos t , a sin t , bt) curvature and torsion are known to be
a = -a22 'b
+
b =a2 b2
+
(where x > Oand 7 > 01'C 0 if a Solving for a and b leads to a = - Y
x* -I- T2'
> Oand b > 0 o r b C 0).
b=-
7
x2
+
T2
If the equilibrium values TO and xeq are inserted into these equations, one obtains the dependences of a and b on the four elastic moduli. The ratio
shows again that k3 must satisfy eq 12 for the helix to be a physical solution. The present fourth-order elastic theory allows for two possible stable states, straight or helical, of an isotropically cylindrical micelle consisting of chiral molecules. In fact, besides the helical fibers of N-octyl-D-gluconamide Fuhrhop and co-workers have observed straight fibers of chiral N-dodecyltartaric acid m ~ n o a m i d e s . The ~ very small diameter of 3.9 nm clearly indicates that the fibers are single cylindrical micelles. It seems possible to adapt the theory to micelles of quadratic cross section. Invariants have then to be constructed from the derivatives of two unit vectors, t, and a director denoting the orientation of the square. External and internal torsion can no longer be separated. The modifications should riot change the basic result that depending on the elastic moduli the micelle prefers to be either straight or helical, but any helices may be irregular if the two torsions, external and internal, are not in phase. Micelles of rectangular cross section should behave like ribbons, i.e. be generally helical. The central formula, eq 9, may be viewed as a fourthorder Landau expansion of the elastic energy. Any expansion can be assumed to be a good approximation only for small enough arguments. Fortunately, the helices of N-octyl-~-gluconamide~ have a very large pitch, b / 2 ~ = 80 nm, so that despite their small radius, a = 2.5 nm, both curvature and torsion are much smaller than the inverse radius of the micelles which is roughly (113.5)nm-'. The requirement of small arguments should therefore be satisfied. Acknowledgment. This work is part of acooperation with J.-H. Fuhrhop which is supported by the Deutsche Forschungsgemeinschaft through SFB 312. I am grateful to him for numerous discussions. (7) Fuhrhop, J.-H.; Demoulin, C.; Rosenberg, J.; Bottcher, Ch. J.A m , Chem. SOC.1990, 112, 2827.
