**Y. Mao, M. Warner**

**May 25, 2000**

Cavendish Laboratory, Madingley Road,

Cambridge, CB3 0HE, UK.

We present a continuum model for a nematic elastomer network
formed in a chiral environment, for instance in the presence of a
chiral solvent. When this environment is removed, the network can
retain some memory of its chiral genesis. We predict the residual
chiral order parameter for a number of possible scenarios, and go
on to examine the robustness (stability) of the imprinted chirality.
We show that a twist-untwist transition can take place,
which determines whether the imprinting has been successful.
A transition is via a coarsening of the helical director pattern and
a lengthening of its pitch. Finally, the effect due to a
subsequent swelling by an achiral solvent, or by a solvent of
differing chirality, is considered.

PACS numbers: 61.30.-v, 61.41.+e,
78.20.Ek

Nematic elastomers combine the properties of a liquid
crystal and those of a conventional rubber. This synergy gives
rise to novel material behaviour, which in turn has stimulated
much research in past years [1]. From a general symmetry
argument, de Gennes [2] first suggested that chirality
may be introduced to such an elastomer by simply forming it in a
chiral solvent. The originally achiral liquid crystalline polymer
would then remember the induced chirality after crosslinking, even
when the solvent is removed or replaced with an achiral one. This
is the case of chiral imprinting, which potentially can open up an
entirely new way of producing materials of specified optical
properties. Chiral imprinting, in principle,
is akin to crosslinking a nematic polymer under an external magnetic or
mechanical field [1,3] where the monodomain state
is also permanently imprinted. Experimentally chiral imprinting
was studied long ago [4] as a function of solvent
exchange. It has been exploited by Yang *et al*,
who used a weak gel to stabilise cholesteric textures or director
uniformity in reflective display devices.
More recently imprinting has been studied [6] as a
function of both solvent removal and temperature.

Another measure of imprinting occurs in intrinsically cholesteric networks. On temperature changes that would cause a substantial pitch variation in a non-crosslinked cholesteric polymer melt, the corresponding network suffers essentially no variation - see for example Fig. 8 of reference [7], where also many other references to cholesteric elastomers are given. In this paper, we analyse chiral imprinting, and predict the retained chiral properties of the elastomer when the initial chiral environment of crosslinking is altered. Gradients of director variation are modeled within continuum Frank nematic elasticity. The nematic elastomer penalty for rotation of the director relative to the solid matrix is described in a fully non-linear, rubber-elastic manner since rotation can be large.

A nematic liquid crystal has a mobile director
,
the gradient of which incurs a Frank energy [8].
For the free energy density of a cholesteric liquid crystal, the twist
term is modified by a pitch wave number, *q*_{0}:

where

When liquid crystalline polymers are crosslinked into a rubber
network, additional constraints on
arise in the form of
director anchoring to the network. Anchoring manifests itself with
an extra energy cost [9] for a uniform director
rotation
relative to a local rotation of the
elastic matrix
.
The energy density is
.
A second term couples the relative director-matrix rotation to
the shearing part of the elastic strain,
, that is
.
For such strains to rotate the director in a cholesteric,
they would have to vary along the helical axis. Elastic
compatibility then introduces prohibitively expensive secondary
shears. Another alternative is an *x*, *y*-dependent
, but this appears
unsupported by experiment [6].
We accordingly ignore
terms.

Consider an initially regular cholesteric helix along the

where

where ' indicates

The two limits of the energy density are (i) the perfectly twisted
cholesteric state
where the
current pitch wavevector is unchanged from *q*_{0}, and (ii) the
untwisted state
,
the additional factor
of
arising from the averaging of over one period. Thus, crudely, we expect the director to be
twisted if

and untwisted otherwise, this balance being tuneable since

To simplify matters, we begin with the following substitutions:

the angle describes the variation away from (or modulation of) the original helical pattern, that is we are now in a rotating frame of reference. Lengths are reduced by the nematic rubber penetration depth , the natural length scale in the problem. The parameter is a non-dimensional measure of the nematic length relative to the chiral pitch, and the condition (3) is equivalent to . Following the remarks below Eq. (3), we expect imprinting to be lost at large and retained at small . Dropping a constant arising from the variable change , the reduced energy (per unit cross-section area perpendicular to the pitch axis) is:

where signifies

The first integral of the Euler-Lagrange equation corresponding to
Eq. (4) , the elliptic equation or often called in the
literature the Sine-Gordon equation, leads to:

where

*The localised limit, **c*^{2}<1*.* Our particle oscillates
between two values of
.
We accordingly
introduce a new variable
in the interval
,
so that
.
Rewriting the derivative
in terms of
and returning it to Eq. (5) reduces
this equation to one of the standard elliptic form:
.
The period of the
oscillatory motion is found to be:

(6) |

Here , and later , are the complete elliptic integrals of the first and second kind respectively. The period

= | (7) |

In order to compare the stability of various states, we require the reduced energy

We now require the energy density in the traveling regime.

*The traveling limit **c*^{2}>1*.* The modulation period down the
original helix corresponds to the time taken for our particle to
travel from one peak to the next in the potential. It is
calculated using Eq. (5):

(9) |

where

= | (10) |

The corresponding energy density,

Combining *g*_{1}(*c*) and *g*_{2}(*c*), Eq.s (8, 11),
we now have the energy density *g*(*c*) for the
entire range of *c*. Fig. 2 illustrates the energy density plots
for three different values of of the chiral strength, described by
the parameter .

We can see that at the transition point

we can minimise the reduced free energy with respect to

The period of
modulation is
in
units of .
For each such period along the helical pitch axis,
increases by ,
the director unwinds once, and
therefore one loses 1 of the
twists imprinted over the interval *T*. The
imprinting efficiency is given by the fractional number of twists
lost:

(13) |

Provided that

For small chiral power
,
the minimising
condition *dg*/*dc*=0 has no solution and the minimum free energy
occurs exactly at *c*=1. The cusp has the energy density
,
and a logarithmically divergent period *T* which
implies *e*_{0}=1. The director largely follows the original helical
twist, the deviation from which never reaches
within a finite
distance. The imprinting is therefore successful.

For chiral power
,
minima are found for
*c*>1. The period is no longer infinite and twists are lost. For
large ,
*i.e.* the nematic penetration depth large
compared to the cholesteric pitch, we can expand the elliptic
function for small *k* to find
.
Alternatively
and we find
a period of
in units of .
Thus for every actual distance of
,
accumulates an increment of .
That is,
,
corresponding eventually to the case of complete unwinding of
the imprinted helical pattern.

In writing equation (2), we have assumed that the
chiral solvent was completely replaced with an achiral one as was
the case in experiment [6]. However, we can trivially
generalise *F* to the case where the current solvent is chiral:

where

which indicates a chiral environment of the same handedness can help preserve the chirality by reducing the value of (negative values of are mathematically equivalent to , if we switch the sign of the modulation). A chiral environment of the other handedness,

Another interesting case is the induction of a cholesteric state
in an initially uniform nematic network upon introducing
a chiral solvent [12]. That is *q*_{0}=0 and
,
where *q* is the chiral pitch wave number due to the chiral solvent.
In this case, *e*_{0} given in Fig. 2 can be interpreted as the
network resistance to imprinting - until the point reaches
there will be no helicity induced in the network
at all, and thereafter it rises with .

Finally, by varying the amount of solvent in the crossed network, we can also tune, *i.e.*
contract or expand, the volume relative to its original:
.
If the added solvent is nematic, we expect from molecular interpretations [13],
for the special case where the local nematic order is preserved
(the nematic order variation due to the solvent or temperature will be examined elsewhere),
but
as it depends on
crosslink density. Finally assuming an isotropic expansion/contraction,
,
we arrive at

Thus swelling can, albeit weakly, increase the parameter in our model and discourage the preservation of the imprinted chirality.

In conclusion, we have proposed a continuum model for chiral imprinting in nematic elastomers. The model predicts the residual chirality when the chiral solvent is removed. A twist-untwist transition emerges from the theory, and an experimental verification of this transition would be a useful test of the theory presented here. We also expect that chiral imprinting will continue to be important for future display devices [5]; we suggest here how such selective reflection may be modified by the degree of swelling, by temperature and by the character of the solvent. Our further work is being focussed on the mechanical properties of networks; we expect an external mechanical field will have substantial influence on the imprinting of chirality.

We thank E. M. Terentjev, R. B. Meyer, and M. E. Cates for useful discussions. The problem was first suggested to
us by H. Finkelmann. YM is grateful to St John's College,
Cambridge for a research fellowship.