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A microscopic model of cholesteric polypeptide solutions
Y.H. Kim
To cite this version:
Y.H. Kim. A microscopic model of cholesteric polypeptide solutions. Journal de Physique, 1982, 43 (3), pp.559-565. �10.1051/jphys:01982004303055900�. �jpa-00209426�
A microscopic model of cholesteric polypeptide solutions
Y. H. Kim (*)
Department of Physics, University of California, Los Angeles, CA 90024, U.S.A.
(Reçu le 13 juin 1980, révisé le 21 septembre 1981, accepté le 30 octobre 1981)
Résumé. 2014 On propose ici un modèle microscopique de cristaux liquides du type polypeptide cholestérique. Les
molécules de polypeptide peuvent avoir des conformations 03B1-helicoïdales. On suppose que de telles molécules ont des dipoles électriques uniformément distribués sur leur grand axe. Il est supposé que les directions des dipôles
sont perpendiculaires aux grands axes et qu’ils tournent uniformément le long des grands axes. On suppose que les molécules ont des interactions stériques à courte distance et des faibles interactions perturbatrices dipolaires à longue portée. L’énergie libre de la phase cholestérique est calculée en utilisant la théorie de perturbation au
deuxième ordre. La période cholestérique et la dépendance à la concentration sont calculées. Les résultats ont l’ordre de grandeur correct et sont en accord qualitatif avec les observations expérimentales.
Abstract 2014 A microscopic model of cholesteric polypeptide liquid crystals is proposed Polypeptide molecules
may form 03B1-helix conformations. It is assumed that such molecules have uniformly distributed electric dipoles along the long axes of molecules. The directions of the dipoles are assumed to be perpendicular to the long axes
and rotate uniformly along the long axes. It is assumed that the mólecules have short range steric interactions and
perturbative long range weak dipole-dipole interactions. The free energy of the cholesteric phase is calculated by
second order perturbation theory. The cholesteric twist and concentration dependence are calculated. The results have the correct order of magnitude and are in qualitative agreement with experimental observations.
Classification Physics Abstracts
34.20B
This is a theoretical model to show how a simple
helical arrangement of electric dipoles along long axes
of rigid linear molecules in solution may produce a
cholesteric phase. It is well known that some poly- peptides at sufficiently high concentration exhibit a cholesteric mesophase [1]. It is assumed that there is a
uniform distribution of dipoles along long axes of the
a-helix molecules. The dipoles are perpendicular to
the axis and rotate uniformly along the axis forming
molecular chirality (Fig. 1). It is assumed that mole- cules have short range steric interactions which favour
a nematic ordering and perturbative long range weak
dipole-dipole interactions.
In figure 1, q is the wave number of molecular chi-
rality. When q > 0, the chirality of a molecular is
right handed, and when q C 0, it is left handed. In
figure 2, a sketch of a cholesteric phase is shown. The
planes A, B, and C are parallel to the xy plane. The
direction of cholesteric twist is parallel to the z axis.
Fig. 1. - A schematic representation of the arrangement of the dipoles on the long axis of a molecule.
Suppose that there is a dipole P1 at the origin. The
molecule which includes PI lies on the plane A and
makes angle (p with x axis (Fig. 3). Also, consider a dipole P2 at R and the molecule which includes that
dipole lies on the plane B. The direction of that mole- cule is perpendicular to yz plane. The angles a and a2
are coordinates of rotational degree of freedom of
molecules with respect to their long axes. The helicity
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01982004303055900
560
Fig. 2. - A schematic representation of a cholesteric phase.
The planes A, B, and C are parallel to the xy plane. The
molecules in the planes are shown as short lines. The z
axis is the direction of cholesteric twist
of a molecule is represented as a2 = qx + fl where
is a phase angle in figure 3. The interaction energy
W12 between P, and P2 is given by
where Pi = pê1, P2 = pê2 and ê1 and ê2 are unit
vectors.
From figures 2 and 3, W 12 is given explicitly by
2 -
Fig. 3. - Pi 1 is the dipole at the origin. P2 is the dipole
at R.
(p is the angle the molecules in the xy plane make with res- pect to the x axis. The column has the cross section of unity
and its centre line is the line D.
The interaction energy Wi of P, with all dipoles
of a molecule in the plane B whose centre of mass is
at (x’, y, z) is given by
where il is the number of dipoles per unit length and
2 1 is the length of a molecule. Next, consider mole- cules which are included in a column of cross section of a2 whose axis coincides with the line D which passes
through (x, y, z) and is parallel to x axis. It is assumed
that the length of a segment of molecules (i.e. polymers)
is o a » and each segment has one dipole (i.e. qa = 1).
In the zeroth order approximation (i.e. in the absence of the perturbation), the distribution of angles
among molecules is uniform in 0 # 2 n. Then,
the interaction energy W of P, with molecules in the column whose angles of rotational degree of freedom
with respect to their long axes are between P and fl + dfl is given by, from equation (3),
where p is the number of molecules per unit volume.
Let fp be the free energy of P1 1 with its rotational
degree of freedom a allowed to fluctuate. According
to second order thermodynamic perturbation theo-
ry [2], ./p is given by
where temperature T is in the unit of energy and W is the average of W over the angles of a and fl with
the zeroth order distribution, fpo the zeroth order term of fp. The use of equation (5) is justified in the
last part of the paper by showing that (W - W )2 T
at room temperature and with other parameters of appropriate values.
From equation (2) - equation (5, it is shown, in Appendix I, that W = 0 and
+ [terms of even powers of qJ (square terms)] (6)
where K1 1 = K I (I q L) is the modified Bessel func- tion [3] with the argument 1 q L. From equation (5)
and equation (6), the free energy fA of PI due to inter-
actions with all molecules in the plane B is given by
where fAO is the sum of the terms of even power of qJ.
According to equation (7), fA is lower if qJ > 0 when q > 0. This means that the cholesteric twist is left handed when the molecular chirality is right handed If q 0, the cholesteric twist is right handed
It is interesting to notice that W. Goossens [4]
reported that there is no direct correlation between the sign and magnitude of the optical activity and the sign and magnitude of cholesteric twist He also stated that [4] the cholesteric phase occurs only with opti- cally active molecules and it seems reasonable to conclude that those molecular properties that are res- ponsible for the optical activity also give rise to the
cholesteric twist. Others [5] found that the sense of the molecular chirality does not dictate the sense of the cholesteric twist T. Samulski and E. Samulski [6]
proposed a theory about the reversal of cholesteric twist of lyotropic polypeptide liquid crystals triggered by a change of certain solvents [5]. In this paper, we showed in equation (7) that our model predicts that
the cholesteric twist and molecular chirality have opposite chiralities (i.e. kq 0). This means that, for example, if the interaction between molecules and solvents produces dipoles on the long axis of a mole-
cule like our model and its chirality, then, change of
some solvents may result in the reversal of cholesteric twist too.
Let us go back to equation (7). The factor sin 2(g)
means that 9 and (p ± n are not distinguished. In
other words, there is no difference of head and tail of
a molecule in our model. For the calculational pur- pose, let us redefine qJ as the angle of directions of molecules in the plane B when molecules in the plane A
are parallel to x axis. This is equivalent to replacing 9
with - 9 in equation (7). Then, the free energy fB
of one molecule at the origin (whose direction of long
axis now coincides with x axis) due to its interactions with the rest of molecules in the entire space is given by
Fig. 4. - f is the free energy density and I k is the magni-
tude of the wave vector of the cholesteric twist
where fBO is the sum of terms of even powers of (p.
In volume a3, there are 2 I r¡a3 p dipoles. Therefore,
the free energy F of molecules in a volume of a3 due to the interactions with rest of molecules in the entire space is given by
where the wave number of cholesteric twist is defined
as kz = 2 (p. The minus sign in front of the second
term of equation (9) is due to the fact that qk 0.
The expression of Fo is obtained from the following arguments : i) Fo should be due to the restoring force
of nematic liquid crystals which tends to keep I k as
small as possible. When k is small (which is the case
in experimental observations [1]), Fo should be pro- portional to k2 with all higher order terms of k 2
eliminated ii) In lyotropic liquid crystals, Fo is essen- tially due to the short range steric interactions bet-
ween hard rods [7]. Therefore, Fo should be propor- tional to N 2 where N is the number of segments of a molecule [8]. From these two considerations, Fo is given by
where v is the coupling constant of rod-rod interac- tions and it has dimensionality of energy. After some
steps (Appendix II), we have
where f =- F/(va’ pN 2 q2), a dimensionless quantity
which corresponds to the free energy F, a = p’ N/4 7rev
with the assumption that a q = 1 (i.e. there is one dipole per segment), and J 1 (I k/q I ç) is the Bessel function of the first kind of order 1. The lower limit
ro I q I of integration is necessary in order to avoid the interaction of a molecule with itself where ro is the radius of the excluded cross section of a molecule.
The magnitude of cholesteric twist I k is obtained by minimizing f with respect to I k 1. This is done nume-
rically using the parameters as
With these values of parameters, we have for the minimum of f.
The results are shown in table I and figure 5.
562
Fig. 5. - The dependence of the cholesteric twist on the concentration of molecules is shown on a log vs. log scale.
p is the concentration of molecules and klq is the ratio
between the wave number of the cholesteric phase and the
wave number of the molecular chirality. k is inversely proportional to the cholesteric pitch.
We find that I k oc p2 and the corresponding choles-
teric pitch s = III k I is s oc p-2. This may be seen
from equation (11) because in the limit of small I klq 1,
As
grows, k
becomes no longer small. Buq
K )(j) falls to zero so fast [3] that there is no significant
contribution to the integration
when k
> 1.q
This dependence of I k I on p is similar to those reported by Robinson et al. [9] and by Uematsu and Uemat-
su [10]. _
Now, let us reexamine the assumption of W 2 T
in order to show the validity of using the perturbation expansion of equation (5). Actually, it is more conve-
nient to reexamine equation (11) instead of equation (5)
since they are equivalent. The value of integration in
equation (11) is essentially in the order of unity because K î(ç) is a rapidly decreasing function of ç. By using
the values of parameters of p, p etc. as those used for table I, it is easily seen that the numerator of the
second term of equation (11) is much smaller than T at room temperature (i.e. up’ - 10-19 erg and T - 10-14 erg). Therefore, the assumption of W ’ T is good with the values of parameters used to obtain the results in table I.
To summarize, we list the following features of the model :
i) The right handed molecular chirality produces
left handed cholesteric twist The left handed chirality produces right handed cholesteric twist.
ii) The cholesteric pitch has the correct-order of
magnitude and the ratio I klq its about 10-2 _ 10-5 for typical values of parameters.
iii) The dependence of cholesteric pitch s on the
concentration of molecules p is s oc p - 2 which is in agreement with experimental observations.
Acknowledgments. - The author thanks Professor
Philip Pincus for his suggestion of this problem and helpful discussions. Also, he wishes to thank Professor E. T. Samulski for his valuable comments. This work
was supported in part by the National Science Foun- dation and by the Office of Naval Research.
Appendix I. - From equation (3)
The following integrations are performed for W
and
Ko(qL) and K1(qL) are modified Bessel functions [3] and the following identities are used :
where a > 0 and the real part of # is greater than zero.
Also, we have
From equations (I.3) and (I.4), we have
Similarly,
From these results, W is given by
564
Therefore,
and
where
- [terms of even power of (p (square terms)] . (1.14)
The factor 1/4 n2 is the normalization constant From equations (5), (1.12) and (I .14), we obtain equation (6).
Appendix II. - If we use polar coordinates r =
IZ 2 +-y 2
and z = r cos 0 and introduce an excluded circular area of ’õ at the origin in order to avoid the interaction of a molecule with itself where ro is the radius of a molecule (i.e. the radius of a long rod), we haveand
Let us introduce a dimensionless quantity j = I q ro. Then, from equations (11. 1) and (II . 2) we have
Since 8 l3 1’ = N’ and f =- F/(va7 pN I q2), we obtain equation (11) from equations (9) and (11.3).
References
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